Structural properties of hierarchically hyperbolic groups

Uni e sidad del Pa´ıs Vasco–Euskal He iko Unibe si a ea
Ph.D. Thesis
S uc u al p ope ies o hie a chically hype bolic
g oups.
B uno Robbio
Supe ised by Ilya V. Kazachko and Ma k F. Hagen
Submi ed on Oc obe 2020
(cc)2020 BRUNO ROBBIO CAMOGLIO (cc by-nc 4.0)
2
Abs ac
The opics o his disse a ion a e amed in he a ea o geome ic g oup heo y, ha is he
s udy o ini ely gene a ed g oups h ough he explo a ion o i s geome ic and opological aspec s.
Mo e p ecisely, we ocus on a class o g oups called hie a chically hype bolic g oups. Hie a chical
hype bolici y is a e y ecen bu powe ul no ion whose goal is o p o ide a uni ying amewo k o
s udy la ge classes o g oups ha ing ea u es eminiscen o non-posi i e and nega i e cu a u e.
We include an in oduc ion o his class o g oups in he i s chap e .
The i s o iginal esul s o his hesis appea in Chap e 2, whe e a numbe o s uc u al esul s on
hie a chically hype bolic spaces a e p o ed. In addi ion, wo no ions a e p esen ed he e: he in e -
sec ion p ope y and conc e eness. These key condi ions a e used in nume ous places h oughou
he es o he hesis and a e c ucial o unde s anding he main esul s ha ollow.
The i s main con ibu ion o he hesis is he es ablishing o a combina ion heo em o he class
o hie a chically hype bolic g oups. We usually e e o a esul as a combina ion heo em on a
class o g oups Ci i p o ides an answe o he ollowing ques ion: Le Gbe a g oup ac ing on
a simplicial ee Twi h e ex and edge s abilize s in C, unde wha condi ions can we conclude
ha he g oup Gis i sel in C? In ou case, he condi ions ha we iden i ied a e he in e sec ion
p ope y and clean con aine s. As an applica ion o his heo em we ob ain ha g aph p oduc s o
hie a chically hype bolic g oups wi h he in e sec ion p ope y and clean con aine s a e hemsel es
hie a chically hype bolic.
In he las chap e o he hesis we ocus on he class o g oups ha ac on a simplicial ee such
ha he e ex s abilize s a e hype bolic and edge s abilize s a e i ually cyclic. We call his
class hype bolic-2-decomposable g oups. We ob ain a cha ac e iza ion o g oups o his ype ha
allows us o p o ide a hie a chical hype bolic s uc u e on hem. Mo e p ecisely, we ob ain ha
a hype bolic-2-decomposable g oup is hie a chically hype bolic i and only i i is balanced. E en
mo e, we show ha his is equi alen o he g oup i sel no con aining non-euclidean Baumslag-
Soli a subg oups. As an immedia e co olla y we ob ain ha ee p oduc s wi h amalgama ion o
hype bolic g oups o e i ually cyclic g oups a e hie a chically hype bolic.
i
ii
Resumen
Los emas de es a esis se enma can en el ´a ea de la eo ´ıa geom´e ica de g upos, que es el es udio de
g upos ini amen e gene ados a a ´es de la explo aci´on de sus aspec os geom´e icos y opol´ogicos.
M´as p ecisamen e, nos cen amos en una clase de g upos denominados g upos je ´a quicamen e
hipe b´olicos. La hipe bolicidad je ´a quica es una noci´on muy ecien e pe o pode osa cuyo obje i o
es p opo ciona un ma co uni icado pa a es udia g andes clases de g upos que ienen ca ac-
e ´ıs icas simila es a cu a u a nega i a y no posi i a. Inclu´ımos una in oducci´on a ´es a clase de
g upos en el p ime cap´ı ulo.
Los p ime os esul ados o iginales de es a esis apa ecen en el cap´ı ulo 2, donde se p ueban una
se ie de esul ados es uc u ales sob e espacios je ´ quicamen e hipe b´olicos. Se p esen an, adem´as,
dos nociones: in e sec ion p ope y y conc e eness. Es as condiciones se u ilizan en a ios luga es
a lo la go del es o de la esis y son c uciales pa a comp ende los p incipales esul ados que siguen.
La p ime a con ibucin p incipal de la esis es el es ablecimien o de un eo ema de combinaci´on
pa a la clase de g upos je ´a quicamen e hipe b´olicos. Po lo gene al, nos e e imos a un esul ado
como un eo ema de combinaci´on en una clase de g upos Csi esponde a la siguien e p egun a:
Sea Gun g upo que ac ´ua sob e un ´a bol simplicial Tcuyos es abilizado es de ´e ices y a is as
pe enecen a C, bajo qu´e condiciones podemos conclui que el g upo Ges ´a en C? En nues o caso,
las condiciones que iden i icamos son in e sec ion p ope y y clean con aine s. Como aplicaci´on de
es e eo ema ob enemos que los p oduc os bajo g a os de g upos je ´a quicamen e hipe b´olicos con
in e sec ion p ope y y clean con aine s son en s´ı mismos je ´a quicamen e hipe b´olicos.
En el ´ul imo cap´ı ulo de la esis nos cen amos en la clase de g upos que ac ´uan sob e un ´a bol
simplicial de mane a que los es abilizado es de a is as son i ualmen e c´ıclicos. Llamamos a
es a clase g upos hype bolic-2-decomposable. El p incipal esul ado de ´es e ´ul imo cap´ı ulo es
una ca ac e izaci´on de g upos de es e ipo que nos pe mi en apo a una es uc u a hipe b´olica
je ´a quica sob e ellos. M´as p ecisamen e, ob enemos que un g upo hype bolic-2-decomposable es
je ´a quicamen e hipe b´olico si y solo si es equilib ado. A´un m´as, mos amos que es o es equi alen e
a que el g upo en s´ı no con enga subg upos de ipo Baumslag-Soli a no equilib ados. Como
co ola io inmedia o ob enemos que los p oduc os lib es amalgamados de g upos hipe b´olicos sob e
g upos i ualmen e c´ıclicos son je ´a quicamen e hipe b´olicos.
Con en s
i
In oduc ion
1 P elimina ies 1
1.1 Geome yo g oups ................................... 1
1.2 Hype bolicg oups .................................... 3
1.2.1 Quasicon exi y.................................. 4
1.3 G aph o g oups and Bass-Se e Theo y . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Rela i ely hype bolic g oups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Hie a chically hype bolic spaces: in oduc ion . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 P ojec ions and coo dina e sys em . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.2 Cons uc ing s uc u es in main examples . . . . . . . . . . . . . . . . . . . 12
1.6 Hie a chically hype bolic spaces: ull de ini ion . . . . . . . . . . . . . . . . . . . . 14
1.7 Hie a chically hype bolic g oups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.1 Hie a chical quasicon exi y and ga e maps . . . . . . . . . . . . . . . . . . . 18
1.7.2 Ga emaps .................................... 20
1.8 P oduc egions...................................... 20
1.9 Cons uc ing examples o hie a chical hype bolici y . . . . . . . . . . . . . . . . . . 23
1.9.1 Hie a chically hype bolic s uc u es on g oups ac ing on ees . . . . . . . . 25
1.9.2 A cha ac e iza ion o hie a chical hype bolici y in hype bolic-2-decomposable
g oups....................................... 28
1.9.3 Ano eon o sion................................. 30
2 S uc u al esul s 33
2.1 In e sec ion p ope y and conc e eness . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 P oo o hemainTheo em ............................... 43
2.3 Mains uc u al esul s.................................. 49
3 A Combina ion heo em 53
3.1 T ees o hie a chically hype bolic spaces . . . . . . . . . . . . . . . . . . . . . . . . 54
iii

i CONTENTS
3.1.1 T ees wi h deco a ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Endowing a ee o HHS wi h an HHS s uc u e . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Cons uc ion o index se . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Hype bolic spaces associa ed o he index se and p ojec ions . . . . . . . . 63
3.2.3 P ojec ions be ween hype bolic spaces . . . . . . . . . . . . . . . . . . . . . 64
3.2.4 P oo o he main heo em . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Applica ions........................................ 78
3.3.1 G aph o hie a chically hype bolic g oups . . . . . . . . . . . . . . . . . . . 79
3.3.2 G aphp oduc s.................................. 86
4 Hype bolic-2-decomposable g oups ha a e HHG 91
4.0.1 Ques ions..................................... 91
4.0.2 Balancedg oups ................................. 92
4.0.3 Con exi y..................................... 96
4.1 Hie a chical hype bolici y o (2-ended)-2-decomposable g oups . . . . . . . . . . . 97
4.1.1 Two-endedg oups ................................ 97
4.1.2 Pulling back hie a chical s uc u es . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.3 Linea ly pa ame izable g aph o g oups . . . . . . . . . . . . . . . . . . . . 99
4.1.4 Cha ac e iza ions o hie a chical hype bolici y . . . . . . . . . . . . . . . . 103
4.2 Hie a chical hype bolici y o hype bolic-2-decomposable g oups . . . . . . . . . . . 106
4.2.1 Commensu abili y and conjugacy g aph . . . . . . . . . . . . . . . . . . . . 109
Bibliog aphy 117
In oduc ion
Hie a chically hype bolic spaces and g oups (HHSs and HHGs) we e in oduced by Beh s ock,
Hagen and Sis o in a se ies o pape s [12, 14]. This is a b oad class ha includes an imp essi e
amoun o spaces and g oups na u ally occu ing om geome ic conside a ions. Mapping class
g oup o su aces; CAT(0)-cube complexes; Teichmulle space wi h he Teichmulle and Weil-
Pe e sson me ic and undamen al g oups o 3-mani olds wi h no Nil no Sol componen a e among
he mos amous objec s admi ing a hie a chical hype bolic s uc u e.
Se e al gene aliza ions o hype bolic g oups ha e been in oduced o e he yea s o desc ibe g oups
o geome ic o igin ha exhibi some no ion o nega i e cu a u e. Rela i e hype bolici y ([22,
37]) eco e s undamen al g oups o 3-mani olds wi h cusps, whe eas mapping class g oups a e
examples o acylind ically hype bolic g oups [69], and aags ( ha is igh -angled A in g oups) a e
among he g oups ac ing p ope ly and cocompac ly on CAT(0) cube complexes, ha is cubulable
g oups [76, 91]. Mo eo e , mapping class g oups a e no ela i ely hype bolic (unless hey a e
al eady hype bolic [8, Theo em 1.2]). The no ion o hie a chical hype bolici y eme ges as a class
ha gene alizes hype bolici y, engul s many o he abo e men ioned g oups and also main ains
many o hei algeb aic ea u es.
Being a hie a chically hype bolic g oup p esen s a wide ange o bo h algeb aic and geome ic
consequences. Some o hese a e a quad a ic isope ime ic inequali y; ini e asymp o ic dimension;
a e sion o he Ti s al e na i e; ank idigi y heo ems and a con olled way in which quasi- la s
a e dis ibu ed in he g oup.
The key insigh o de ine hie a chically hype bolic g oups is he axioma iza ion o he Masu -
Minsky machine y de eloped o mapping class g oups o gene al g oups. E o s in his di ec ion
a e no a no el y in ce ain classes o g oups. Fo ins ance, in [80] he au ho p esen s a way o
cha ac e izing ela i e hype bolici y in e ms o p ojec ions simila o ha o subsu ace p ojec ions
in he cu e g aph and de elops a dis ance o mula. Mo eo e , in [46], he au ho in oduces he
con ac g aph o cubical g oups, an analog o he con ac g aph o cube complexes.
A hie a chical hype bolic s uc u e on a geodesic me ic space Xis composed o he ollowing da a:
1. An index se S;
2. a collec ion o δ-hype bolic spaces;
3. a collec ion o p ojec ions πV:XÑCVuVPS.
This da a mus sa is y a se o axioms. The ull de ini ion is included in Sec ion 1.6.
i INTRODUCTION
O ganiza ion o he hesis
This hesis is di ided in o ou chap e s. Chap e 1 is exposi o y and ecollec s basic concep s on
coa se geome y and geome y o g oups. The main opics included in his chap e a e hype bolic
g oups (Sec ion 1.2), Bass-Se e heo y (Sec ion 1.3), and ela i ely hype bolic g oups (Sec ion
1.4). We also gi e an in oduc ion o he de ini ion o hie a chically hype bolic spaces and g oups
(Sec ion 1.6), which a e he main objec o s udy h oughou he es o he wo k. The inal sec ion
o his i s chap e (Sec ion 1.9) deals wi h examples o hie a chically hype bolic g oups, and is
in ended as an in oduc ion and mo i a ion o he o iginal wo k ha is p esen ed in his hesis.
The eade ha is well- e sed in hie a chical hype bolici y may wish o s a he eading o he
hesis in his sec ion. The emaining chap e s comp ise he o iginal con ibu ions o he au ho ,
wi h Chap e s 2 and 3 being pa o a join wo k wi h Fede ico Be lai ([15]) and Chap e 4 pa
o a join wo k wi h Da ide Sp iano ([71]).
Chap e 2 concen a es on s uc u al p ope ies o hie a chically hype bolic spaces and hie omo -
phisms (i.e mo phisms in he class o HHGs). We in oduce he no ions o in e sec ion p ope y, o
ε-suppo , and o conc e eness o a hie a chically hype bolic space (see De ini ion 2.1.1, De ini ion
2.1.6, and De ini ion 2.1.10). All o hese will be necessa y o Chap e 3. The main heo em o his
chap e is Theo em 2.2.1 which is hen used in he p oo s o Theo em 2.3.3 and Lemma 2.3.4. These
esul s will be applied epea edly in Chap e 3, which is de o ed o he p oo o Theo em 3.0.1.
Chap e 3 we p esen and p o e a combina ion heo em on hie a chically hype bolic spaces (The-
o em 3.0.1). Sec ion 3.1 is conce ned wi h ees o hie a chically hype bolic spaces, which is an
ex ension o he no ion o ees o spaces o he class o HHSs. In Subsec ion 3.1.1 we in oduce a
ick, which we call he deco a ion o a ee o hie a chically hype bolic spaces T, which is unda-
men al o ou app oach o p o e Theo em 3.0.1. To a ee o HHSs Twe associa e a o al space
XpTq ha , in Sec ion 3.2, we p o e ha can be endowed wi h a hie a chical hype bolic s uc u e.
Sec ion 3.3 is conce ned wi h wo applica ions o Theo em 3.0.1. The i s one, Co olla y 3.3.1 is
a combina ion heo em o hie a chically hype bolic g oups. As a byp oduc o Theo em 3.3.7, we
ex end he esul s o [2] o show ha clean con aine s a e no only p ese ed by aking ee and
di ec p oduc s, bu also by g aph p oduc s.
Chap e 4 is de o ed o he applica ion o he combina ion heo em de eloped in he p e ious
chap e o g oups ha spli as g aphs o g oups wi h hype bolic e ex g oups and 2-ended edge
subg oups. To abb e ia e, i Pis a p ope y o a g oup, we say ha a g oup is P-2-decomposable
i i spli s as a g aph o g oups wi h 2-ended edge g oups and e ex g oups sa is ying p ope y
P. The main esul o his chap e is ha a hype bolic-2-decomposable g oup has a hie a chical
hype bolic s uc u e i and only i i is balanced (Co olla y 4.2.16). I he g oup is u he assumed
o be i ually o sion- ee, we ob ain ha a hype bolic-2-decomposable g oup is hie a chically
hype bolic i and only i con ains no non-euclidean Baumslag-Soli a subg oup (Co olla y4.2.15).
In Sec ion 4.1 we in oduce he no ion o linea pa ame iza ion (De ini ion 4.1.11) on (2-ended)-2-
decomposable g oups and use his o p o e he main esul o his chap e o ha class (Theo ems
4.1.25 and 4.1.24). In Sec ion 4.2 we p o e Theo em 4.2.2, which allows us o ex end he esul s
de eloped in Sec ion 4.1.4 o he mo e gene al class o hype bolic-2-decomposable g oups.
Chap e 1
P elimina ies
This chap e is mean as an in oduc ion o he main aspec s o geome ic g oup heo y, aimed a
p esen ing hie a chically hype bolic spaces and g oups and how hey i in o he a ea. We begin
by ecalling he basic de ini ions and objec s ha will appea h oughou he sec ion.
1.1 Geome y o g oups
De ini ion 1.1.1. Le Gbe a g oup and le Xbe a me ic space such ha Gac s on Xby
isome ies. We say ha he ac ion is
1. p ope ly discon inuous i o all compac KĎX, | gPG|gK XK‰∅u| ă 8.
2. cocompac i X{Gis compac in he quo ien opology.
3. The me ic space Xis p ope i closed balls a e compac .
We o en use he abb e ia ion o geome ic ac ion o e e o a p ope ly discon inuous and cocom-
pac ac ion o Gon X. F om now on, when we say ha a g oup Gac s on a me ic space Xwe
assume ha he ac ion is by isome ies, unless o he wise s a ed.
De ini ion 1.1.2 (Cayley g aph). I Gis a g oup gene a ed by a ini e se S“ s1, . . . , snuwe
associa e a g aph X o he pai pG, Sqwhe e he unde lying e ex se is Gand wo elemen s g, h
a e a dis ance one in Xi and only i g´1hbelongs in S. This g aph Xis known as he Cayley
g aph o Gwi h espec o S.
Associa ing a Cayley g aph o a ini ely gene a ed g oup can be iewed as a p ocess ha con e s
g oups o me ic spaces. Fu he , i is s aigh o wa d o check ha a ini ely gene a ed g oup ac s
geome ically on any o i s Cayley g aphs. We use X“CaypG, Sq o deno e he Cayley g aph o
a g oup wi h espec o a gene a ing se S.
A c ucial obse a ion says ha he la ge-scale s uc u e o a Cayley g aph does no depend on he
choice o gene a ing se . This obse a ion is usually e e ed o as he Milno -S a c lemma:
1
8CHAPTER 1. PRELIMINARIES
he subwo d mus appea in “p 11 2. . . sq`u0 1
e1u1. . . m
emum˘‰. Since uwas assumed o be
educed and 1, . . . , sis a sho es pa h, he subwo d mus be su0 1
e1, whe e u0“φ `
spzq o
some zPG s. Then eplace su0 1
e1by φ ´
spzq, and pe o m he symme ic change on he o he
side o he x. No e ha his p ocess educes he leng h o he pa h 1, . . . , sby one. In pa icula ,
i has o e mina e.
So, assume ha no educ ion can be pe o med in pu ““p 11 2. . . sq`u0 1
e1u1. . . m
emum˘‰. I
pu “h0PGw, and hence x, y PGwa e conjuga e in Gw, we a e done. So suppose his is no he
case. We need o ha e ha umxu´1
m“φ`
empzq o some zPGem. Subs i u e m
emumxu´1
m ´m
emwi h
m´1
emφe´
mpzq ´m`1
em. I mą1, add a pa h con ained in he spanning ee and epea he p ocess
using he no mal o m heo em again, un il we ob ain a educ ion o he o m
m´1
em´1um´1Z0u´1
m´1 ´m´1
em´1,
o some Z0Pφe´
mpGemq. Again, we mus ha e um´1Z0u´1
m´1Pφe`
m´1pGem´1, ha is o say,
um´1φe´
mpGemqu´1
m´1Xφem´1pGem´1q‰ 1u. P oceeding as abo e, we ge he claim o each ui.
Whene e we a e wo king on a g aph o g oups, i is o en he case ha we a e in e es ed in
s udying a subg aph o g oups. Fo ha we adop he ollowing no a ion.
No a ion. Le Gbe a g aph o g oups and Γ i s unde lying g aph. I Λ ĎΓ is a connec ed
subg aph, hen we can de ine he subg aph o g oups G|Λ, whe e he unde lying g aph is Λ, e e y
e ex and edge in Λ has he same associa ed g oups as in Gand he maximal sub ee o Γ is an
ex ension o he maximal sub ee o Λ.
We call G|Λ he subg aph o g oups spanned by Λ.
Lemma 1.3.11. Le Gbe a g aph o g oups and le ΛĎΓbe a subg aph. Le T1ĎΛbe a spanning
ee o Λsuch ha T1can be ex ended o he spanning ee Tin Γ. Then, he e exis s a g oup
injec ion π1pG|Λ, T 1qãÑπ1pG, Tq.
Rema k 1.3.12. 1. I Γ consis s o a single e ex and a single edge e, hen π1pGqis isomo -
phic o he HNN ex ension G ˚φe.
2. I Γ consis s o wo e ices , w and a single edge ejoining hem, hen π1pGqis isomo phic
o he ee p oduc wi h amalgama ion G ˚GeGw.
3. Whene e Γ is a ee, we will call π1pGqa ee p oduc .
De ini ion 1.3.13. We say ha a g oup Gspli s non- i ially i he e exis s a g aph o g oups G
such ha G–π1pGqand such ha Gis no isomo phic o G o Ge o any PVpΓqand ePEpΓq.
We now ecall he undamen al heo em ela ing spli ings o a g oup wi h g oups ac ing on ees.
This is also known as he undamen al Bass-Se e heo em.

1.4. RELATIVELY HYPERBOLIC GROUPS 9
Theo em 1.3.14. Le Gbe a g oup ha spli s non- i ially as G–π1pGq. Then, he e exis s
a ee Ton which Gac s wi hou edge in e sion such ha he ac o g aph T{Gis equal o ΓG.
Mo eo e , he s abilize s o e ices and edges o his ac ion a e conjuga e o e ex and edge g oups
in G espec i ely.
P oo . See, o ins ance, [19, Theo em 12.1] and [19, Theo em 15.1].
No e ha he ee Tdepends on he spli ing o he g oup G. Con e sely, he spli ing o a g oup
Gis de e mined in e ms o bo h he ee on which Gac s and he ac ion.
De ini ion 1.3.15. We call he ee Tassocia ed o a spli ing Go G he Bass-Se e ee.
1.4 Rela i ely hype bolic g oups
As al eady s essed by G omo , some na u al g oups o geome ic o igin do no i in o he hy-
pe bolici y pic u e: Kleinian g oups and undamen al g oups o 3-mani olds wi h cusps a e exam-
ples o his ac . He also no iced ha e en spaces which a e no hype bolic may p esen some
hype bolic-like ea u es in i s geome y. Mo e p ecisely, in [44], he desc ibes a amily o spaces
whe e he absence o hype bolici y is es ic ed o an isola ed ini e collec ion o subg oups. In
a g oup heo e ical language, hese a e g oups whe e he Cayley g aph is hype bolic ou side o a
ini e collec ion o subg oups. These g oups a e known as ela i ely hype bolic g oups, a class ha
gene alizes hype bolic g oups.
Rela i e hype bolici y was o mally in oduced independen ly by B. Fa b and B. Bowdi ch in
[22,37]. E e since, ela i ely hype bolic g oups has been ex ensi ely s udied and shown o be an
ex emely ich objec o analyse om mul iple poin s o iew. To name a ew, ela i ely hype bolic
g oups ha e been s udied in ela ion wi h algo i hmic p ope ies ([68]); asymp o ic cones ([34]);
and quasi- la s ([27]). Mo eo e , a cha ac e iza ion o ela i e hype bolici y in e ms o p ojec ions
has been de eloped in [80].
The e a e mul iple de ini ions o ela i ely hype bolic g oups in he li e a u e (see, o ins ance
[22,37,44]). In his chap e we include he one due o Bowdi ch in [22].
De ini ion 1.4.1. Le Gbe a ini ely gene a ed g oup and le H1, H2, . . . , Hkbe a subg oup o
G. We say ha Gis hype bolic ela i e o H1, . . . , Hki Gac s on a hype bolic g aph Xwi h he
ollowing condi ions:
1. The numbe o o bi s o edges is ini e;
2. ini e edge s abilize s;
3. e ex s abilize s a e ei he ini e o conjuga e o some Hi;
4. he g aph Xis ine: o e e y nPNand any edge o Xis con ained in ini ely many ci cui s
o leng h n. He e, by ci cui we mean a cycle wi hou sel -in e sec ion).
10 CHAPTER 1. PRELIMINARIES
Figu e 1.1: Coned-o Cayley g aph o Gwi h espec o H.
We call a pe iphe al subg oup o each one o he subg oups Hi.
Examples/P ope ies 1.4.2. 1. I H1, H2a e hype bolic g oups and Fis a common ini e
subg oup hen G“H1˚FH2is hype bolic ela i e o H1, H2u. Indeed, he ac ion o Gon
X he Bass-Se e ee co esponding o H1˚FH2sa is ies he condi ions o De ini ion 1.4.1;
2. The g oup Z2“ xa, b | a, bsy is weakly hype bolic wi h espec o xaybu i is no hype bolic
ela i e o i . Indeed, i ha we e he case, hen xaywould s abilize a e ex in Xand xayb
would s abilize a e ex wjoined by an edge o . Thus, xay X xayb“ xaywould s abilize
ha edge. This con adic s condi ion 2 o De ini ion 1.4.1
3. I Gis hype bolic ela i e o a subg oup H hen His almos malno mal in G(i.e |HgXH|ă8
o e e y gPGzH).
4. Le Gbe hype bolic ela i e o a subg oup HďG. I His hype bolic, hen Gis hype bolic.
A use ul cons uc ion when s udying ela i e hype bolici y is he coning-o o a g oup wi h e-
spec o a collec ion o subg oups. This will be pa icula ly help ul when we conside he ela i e
hype bolici y in e ms o he associa ed Cayley g aph i sel ins ead o an abs ac g aph.
De ini ion 1.4.3. [Coned-o Cayley g aph] Le Gbe a ini ely gene a ed g oup and le Hbe
a ini ely gene a ed subg oup o G. Fix a se o gene a o s So G. In he Cayley g aph CaypG, Sq
add a e ex pgHq o each le cose gH o H, and connec pgHqwi h each xPgH by an edge o
leng h 1{2. The ob ained g aph y
CaypG, Sqis called a coned-o g aph o Gwi h espec o H. We
gi e his g aph he pa h me ic. We say ha Gis weakly hype bolic ela i e o Hi y
CaypG, Sqis
aδ-hype bolic me ic space o some δas in De ini ion 1.2.2. No e ha y
CaypG, Sqis no a p ope
me ic space, as closed balls a e no necessa ily compac .
1.5. HIERARCHICALLY HYPERBOLIC SPACES: INTRODUCTION 11
Rema k 1.4.4. I is easy o see ha y
CaypG, Sqis quasi-isome ic o he g aph ob ained om
CaypG, Sqby collapsing each le cose o H o a poin . Howe e , he coned-o Cayley g aph
y
CaypG, Sqwi h espec o His qui e di e en om he g aph CaypG, Sq{Hob ained om quo i-
en ing he ac ion o Hon CaypG, Sq. This is due o he di e ence be ween le and igh cose s o
Hin G. I His no mal in G hen y
CaypG, Sqand CaypG, Sq{Ha e quasi-isome ic.
Lemma 1.4.5. I Gis hype bolic ela i e o a collec ion P hen he hype bolic g aph Xcan be
aken o be he coned-o Cayley g aph o Gwi h espec o P.
Las ly, we include wo esul s on ela i e hype bolic g oups ha an icipa e much o he ollowing
chap e .
Lemma 1.4.6. [P ojec ions][34, Lemma 4.11] Le Gbe a g oup hype bolic ela i e o a ini e
collec ion o subg oups H1, . . . , Hku. I Pis he se o le cose s o pe iphe als in G. Fo each
PPP he closes -poin p ojec ion πP:GÑPis a coa sely Lipschi z map.
Theo em 1.4.7. The e exis s s0so ha o e e y sěs0 he e exis s K, C so ha o e e y
x, y PG
dpx, yq —pK,Cqÿ
PPP dpπPpxq, πPpyqquus`dp
Gpx, yq.
1.5 Hie a chically hype bolic spaces: in oduc ion
Despi e i s success, ela i e hype bolic g oups a e a om comple ing he pic u e o g oups wi h
hype bolic-like ea u es. Pe haps he mos well-known e idence o his ac a e Mapping class
g oups o su aces. Indeed, i has been shown in [6, 8] ha mapping class g oup o a su ace o
complexi y a leas one can ne e be hype bolic ela i e o any collec ion o ini ely gene a ed
subg oups. Howe e , he powe ul Masu -Minsky machine y ([63,64]) de eloped o hese g oups
is a clea indica i e o he mani es a ion o hype bolici y in i . The e o e, one is b ough o ind a
se o p ope ies ha would gene alize hype bolici y, include mapping class g oups, and s ill ha e
s ong algeb aic consequences o g oups sa is ying hem.
These condi ions ha e been iden i ied by Beh s ock, Hagen, and Sis o, who isola ed he no ions o
hie a chically hype bolic spaces and o hie a chically hype bolic g oups [12,14]. Again, he geome ic
app oach ha is unde aken e lec s in o s ong algeb aic and asymp o ic p ope ies: hie a chically
hype bolic g oups a e ini ely p esen ed [14, Co olla y 7.5], hey sa is y a quad a ic isope ime ic
inequali y [14, Co olla y 7.5], hey a e coa se median [14, Theo em 7.3], and hey ha e ini e
asymp o ic dimension [10].
The de ini ion o hie a chically hype bolic spaces is qui e echnical and leng hy. Thus, be o e we
p esen he ull de ini ion we would like o de o e some space o p ope ly mo i a e and in oduce
e e y signi ican aspec o his class. The emphasis o his sec ion is pu on a heu is ic app oach
o he cons uc ion o hie a chical hype bolic s uc u es a he han a echnical o e iew o he
heo y. The expe ienced eade may wish o skip his sec ion.
12 CHAPTER 1. PRELIMINARIES
1.5.1 P ojec ions and coo dina e sys em
A hie a chical hype bolic s uc u e on a geodesic me ic space Xconsis s o he ollowing da a:
1. A collec ion o δ-hype bolic spaces CVu;
2. a se S ha indexes he a ious hype bolic spaces;
3. o e e y VPS, a pK, Kq-coa sely Lipschi z map πV:XÑCV.
The se o indices along wi h he a ious hype bolic spaces endow Xwi h a coo dina e sys em
ha allows o in es iga e he geome ic aspec s o Xby means o i s p ojec ions. Following his
spi i , a hie a chically hype bolic space can be oughly hough o as a me ic space ha can be
decomposed in o building blocks ha a e hype bolic me ic spaces. The mos basic example o a
space wi h his cha ac e is ics is R2, as i can clea ly be decomposed as a di ec p oduc o wo
in ini e lines.
The de ining s uc u e o a hie a chically hype bolic space also con ains h ee ela ions ha encode
how do a ious elemen s in he index se ela e o each o he . These a e called nes ing (deno ed
by Ď); ans e sali y (deno ed by &) and o hogonali y (deno ed by K). Each one o his ela ions
impose condi ions in which he way he hype bolic building blocks i in X.
1.5.2 Cons uc ing s uc u es in main examples
He e we desc ibe he hie a chical hype bolic s uc u e in di e en classes o g oups.
Righ -angled A in g oups Le Γ be a simplicial g aph. We ecall ha he Righ -angled A in
g oup associa ed o Γ is de ined as he g oup gi en by he p esen a ion
AΓ“ xVpΓq| , ws “ 1ô , wu P EpΓqy.
The space CaypAΓqcan be endowed wi h a hie a chically hype bolic s uc u e as ollows.
(Index se ) Le PΓbe he collec ion o all ull subg aphs o Γ. Fo each Λ PPΓwe say ha
wo cose s gAΛ, hAΛa e pa allel i gh´1, AΛs “ 1. No e ha pa allelism de ines an equi alence
ela ion on he se o cose s o AΛ|ΛPPΓu. We use gΛs o deno e he pa allelism class o he
cose gAΛ o each Λ PPΓ. We se he index se S o be gΛs | ΛPPΓ, g PAΓu.
(Hype bolic spaces) To each gΛs P Swe associa e he hype bolic space C gΛsde ined as gp
AΛ,
whe e p
AΛis he Cayley g aph o AΛwi h SΛ“VpΓqY AΛ1ăAΛ|Λ1ĹΛuas gene a ing se .
Theo em 1.5.1. [12] The space C gAΛs “ gp
AΛis quasi-isome ic o a ee, in pa icula i is
hype bolic.
(P ojec ions) Fo each gΛs P Swe associa e he p ojec ion πΛ:AΓÑC gΛsas he composi ion
ι˝pΛ. He e, pΛdeno es he closes -poin p ojec ion on o gAΛin he Cayley g aph o AΓwi h he
s anda d gene a ing se and ιis he inclusion CaypAΛ;VpΛqq Ñ CaypAΛ;SΛq.
1.5. HIERARCHICALLY HYPERBOLIC SPACES: INTRODUCTION 13
G aph o mul icu es
We would now like o ou line he hie a chical hype bolic s uc u e on a g aph o mul icu e. Le
us i s ecall some no ions.
Le S“Sg,n deno e he connec ed, o ien ed su ace o genus gwi h npunc u es. The complex
o cu es CSassocia ed o Swas o iginally in oduced by Ha ey [50]. I is de ined as a complex
whe e he 1-skele on is gi en by he ollowing:
1. Ve ices: The e is one e ex o each iso opy class o essen ial simple closed cu e in S.
2. Edges: The e is an edge be ween pai o e ices in CSwhene e he co esponding iso opy
class o cu es can be ealized disjoin ly.
We assume ha e e y edge in CShas leng h one, making i a me ic space. This means ha i α, β
a e cu es in Ssuch ha dCSp αs, βsq “ n hen he e exis cu es α“α1, . . . , αn“βsuch ha
αisand αi`1scan be ealized disjoin ly o e e y i. While he mapping class g oup o su aces
a e almos ne e hype bolic, he ollowing g oundb eaking esul by Masu and Minsky e idences
a connec ion be ween he mapping class g oup o a su ace and nega i e cu a u e.
Theo em 1.5.2. [63] The e exis s δsuch ha CSis δ-hype bolic, whe e δdepends on S.
We now ecall an impo an ool de eloped by Masu and Minsky. Fo any subsu ace S1o S
we de ine he subsu ace p ojec ion map πS1:CSÑ2CS1as ollows. Le αbe a cu e ealized
in minimal posi ion wi h BSS1( ha is o say, he numbe o poin s in he in e sec ion αXBS1is
minimal in e ms o iso opy). I αis con ained in S1, we de ine πS1pαqas α. I αis disjoin om S1,
we de ine πS1pαqas ∅. O he wise, o each a c ωo in e sec ion o αwi h S1, we ake he bounda y
componen o a small egula neighbou hood o ωYBSS1which a e non-pe iphe al in S1. Then,
we se πS1pαqas he union o hese cu es o e all such ω.
The abo e p ojec ion sys em can be ex ended o a ious ypes o so-called g aphs o mul icu es
in S. A g aph o mul icu es is de ined as a g aph associa ed o a su ace whe e each e ex
co esponds o a collec ion o iso opy class o cu es in S. This no ion ex ends he one o cu e
g aph o a su ace and, o e he pas decades i has a ac ed signi ican a en ion. In [88] he
au ho shows ha a wide ange o examples o cu es o his ype a e hie a chical hype bolic.
We now ocus on he hie a chically hype bolic s uc u e on a speci ic g aph o mul icu es called
he pan s decomposi ion g aph, which we deno e by GpSq. Each e ex in GpSqco esponds o a
mul icu e on S ha de ines a pan decomposi ion. Two e ices , w in he pan s decomposi ion
g aph a e joined by an edge i one o he cu es α in can be eplaced by a cu e αwin w.
(Index se ) We de ine he index se Sas he collec ion o iso opy classes o all possible subsu aces
o S.
(Hype bolic spaces) We associa e o e e y S1PS he cu e g aph CS1.
(P ojec ions) Fo each SPSwe associa e he map πS:GpSq Ñ CSde ined as he subsu ace
p ojec ion desc ibed abo e.
Fo an explici p oo o he hie a chical hype bolici y o many g aphs o mul icu es we e e o
[88].

14 CHAPTER 1. PRELIMINARIES
The wo examples abo e illus a e one o he mos ema kable ea u es in he heo y o hie a chi-
cally hype bolic spaces: i is a class o spaces ha engul s a ious objec s ha seem o be inhe en ly
di e en om a geome ic iewpoin . Fo ins ance, a Cayley g aph o a igh -angled A in g oup
is an example o a CAT(0)-cube complex, whe eas he mapping class g oup o a su ace is almos
ne e a CAT(0) me ic space ([25,55])).
G oups hype bolic ela i e o hie a chically hype bolic g oups
I Gis a g oup which is hype bolic ela i e o a collec ion o hie a chically hype bolic g oups
pHi,SHiqun
i“1 hen CaypGqcan be endowed wi h a hie a chically hype bolic g oup s uc u e.
Fo each i“1...,n and each le cose o Hiin G, ix a ep esen a i e gHi. Le gSibe a copy o
Si. Le p
Gbe he hype bolic space ob ained by coning-o Gwi h espec o he pe iphe als Hiu.
(Index se ) We de ine he index se as S“ p
GuYŮgPgŮiSgHi.
(Hype bolic spaces) Fo each copy gV o an elemen VPSHiwe associa e a copy o CVas
CgV .
(P ojec ions) πp
G:GÑp
Gis he inclusion, which is coa sely su jec i e and hence has quasicon ex
image. Fo each UPSgHi, le ggHi:GÑgHibe he closes -poin p ojec ion on o gHiand le
πG
U“πHi
U˝ggHi, o ex end he domain o πU om gHi o G. Since each πHi
Uwas coa sely Lipschi z
on CUwi h quasicon ex image, and he closes -poin p ojec ion in Gis uni o mly coa sely Lipschi z
(Lemma 1.4.6), he p ojec ion πG
Uis uni o mly coa sely Lipschi z and has quasicon ex image.
1.6 Hie a chically hype bolic spaces: ull de ini ion
The de ini ion o Hie a chically hype bolic spaces and g oups can be ound in [12] and [14]. I is
also wo h men ioning ha in [82] a e y accessible and iendly in oduc ion can be ound.
We now p esen he de ini ion o hie a chically hype bolic spaces and g oups in i s ull gene ali y
and subsequen ly examine he a ious ing edien s in de ail.
De ini ion 1.6.1. Aq-quasigeodesic me ic space pX, dXqis hie a chically hype bolic i he e exis
δě0, an index se S, and a se CW|WPSuo δ-hype bolic spaces pCU, dUq, such ha he
ollowing condi ions a e sa is ied:
1. (P ojec ions) The e is a se πW:XÑ2CW|WPSuo p ojec ions ha send poin s in X
o se s o diame e bounded by some ξě0 in he hype bolic spaces CWPS. Mo eo e , he e
exis s Kso ha all WPS, he coa se map πWis pK, Kq-coa sely lipschi z and πWpXq1is
K-quasicon ex in CW.
2. (Nes ing) The index se Sis equipped wi h a pa ial o de Ďcalled nes ing, and ei he S
is emp y o i con ains a unique Ď-maximal elemen . When VĎW,Vis nes ed in o W.
Fo each WPS,WĎW, and wi h SWwe deno e he se o all VPS ha a e nes ed in
W. Fo all V, W PSsuch ha VĹW he e is a subse ρV
WĎCWwi h diame e a mos ξ,
and a map ρW
V:CWÑ2CV.
1I AĎX, by πUpAqwe mean ŤaPAπUpaq.
1.6. HIERARCHICALLY HYPERBOLIC SPACES: FULL DEFINITION 15
3. (O hogonali y) The se Shas a symme ic and an i e lexi e ela ion Kcalled o hogonal-
i y. Whene e VĎWand WKU, hen VKUas well. Fo each ZPSand each UPSZ
o which VPSZ|VKUu‰H, he e exis s con Z
KUPSZz Zusuch ha whene e VKU
and VĎZ, hen VĎcon Z
KU.
4. (T ans e sali y and Consis ency) I V, W PSa e no o hogonal and nei he is nes ed
in o he o he , hen hey a e ans e se: V&W. The e exis s κ0ě0 such ha i V&W, hen
he e a e se s ρV
WĎCWand ρW
VĎCV, each o diame e a mos ξ, sa is ying
min dWpπWpxq, ρV
Wq, dVpπVpxq, ρW
Vq(ďκ0,@xPX.
Mo eo e , o VĎWand o all xPXwe ha e ha
min dWpπWpxq, ρV
Wq,diamCVpπVpxqYρW
VpπWpxqqq(ďκ0.
In he case o VĎW, we ha e ha dUpρV
U, ρW
Uq ď κ0whene e UPSis such ha ei he
WĹU, o W&Uand UMV.
5. (Fini e complexi y) The e is a na u al numbe ně0, he complexi y o Xwi h espec
o S, such ha any se o pai wise Ď-compa able elemen s o Shas ca dinali y a mos n.
6. (La ge links) The e exis λě1 and Eěmax ξ, κ0usuch ha , gi en any WPSand
x, x1PX, he e exis s Tiui“1,..., NuĂSWz Wusuch ha o all TPSWz Wuei he TPSTi
o some i, o dTpπTpxq, πTpx1qq ă E, whe e N“λdWpπWpxq, πWpx1qq ` λ. Mo eo e ,
dWpπWpxq, ρTi
Wq ď N o all i.
7. (Bounded geodesic image) Fo all WPS, all VPSWz Wuand all geodesics γo CW,
ei he diamCVpρW
Vpγqq ď Eo γXNEpρV
Wq‰H.
8. (Pa ial ealiza ion) The e is a cons an αsa is ying: le Vjube a amily o pai wise
o hogonal elemen s o S, ad le pjPπVjpXq Ď CVj. Then he e exis s xPXsuch ha
•dVj`πVjpxq, pj˘ďα o all j;
• o all jand all VPSsuch ha V&Vjo VjĎVwe ha e dVpπVpxq, ρVj
Vq ď α.
9. (Uniqueness) Fo each κě0 he e exis s θu“θupκqsuch ha i x, y PXand dpx, yq ě θu,
hen he e exis s VPSsuch ha dVpx, yq ě κ.
The inequali ies o he ou h axiom a e called consis ency inequali ies.
Rema k 1.6.2. The elemen con Z
KUappea ing in Axiom (3) o De ini ion 1.6.1 is called he
o hogonal con aine (o he con aine o he o hogonal complemen ) o Uin Z. I Zis he
16 CHAPTER 1. PRELIMINARIES
Ď-maximal elemen o S, hen we migh supp ess i om he no a ion, w i e con KUand call i
highe con aine . I Zis no he Ď-maximal, hen we will alk abou lowe con aine s.
A hie a chically hype bolic space has clean con aine s i UKcon Z
KU o all U, Z PS, as o iginally
de ined in [2, De ini ion 3.4].
Fo a hie a chically hype bolic space pX,Sqand a subse UĎS, we de ine
(1.1) UK:“ VPS|VKU o e e y UPUu.
We usually use he uple pX,Sq o deno e a hie a chically hype bolic space, whe e Xis a me ic
space and Sis he collec ion o δ-hype bolic spaces. Be o e di ing deepe in o he heo y, le us
show a ew basic examples o hie a chically hype bolic spaces.
Examples/P ope ies 1.6.3. 1. I Xis hype bolic, hen pX, Xuq is a hie a chically hype -
bolic space s uc u e, whe e he p ojec ion πXis idX;
2. I Z2“ xa, b | a, bsy hen we can endow Z2has a hie a chically hype bolic s uc u e whe e
he associa ed hype bolic spaces a e he cose s o he subg oups xay,xbyand he coned-o
space S“y
CaypZ2qwi h espec o xayand xby. The ollowing ela ions a e imposed:
•xay K xby
•xayĎSand xbyĎS
3. I pX1,S1q,pX2,S2qa e HHS, hen pX1ˆX2,S1YS2qis a hie a chically hype bolic space;
4. [14, Theo em 9.1] Le Gbe a g oup hype bolic ela i e o a ini e collec ion Po pe iphe al
subg oups. I each PPPis a hie a chically hype bolic g oup hen Gis a hie a chically
hype bolic g oup.
Rema k 1.6.4. By [14, Rema k 1.3], he p ojec ions πUo a hie a chically hype bolic space
pX,Sqcan always be assumed o be uni o mly coa sely su jec i e. Wi hou loss o gene ali y, we
will always assume his.
Rema k 1.6.5. I pX,Sqis a hie a chically hype bolic space and he e exis s a me ic space Y
and a quasi-isome y q:XÑY hen Ycan be endowed wi h he hie a chical hype bolic space
s uc u e pY,Sq. Indeed, o do so i is enough o keep e e y elemen in he index se Sand de ine
p ojec ions o e e y WPSas πW˝q, whe e qdeno es a quasi-in e se o q.
De ini ion 1.6.6 (Hie omo phism). Le pX,Sqand pX1,S1qbe hie a chically hype bolic spaces.
Ahie omo phism is a iple φ“`φ, φ♦, φ˚
UuUPS˘, whe e φ:XÑX1is a map, φ♦:SÑS1is an
injec i e map ha p ese es nes ing, ans e sali y and o hogonali y, and, o e e y UPS, he
maps φ˚
U:CUÑCφ♦pUqa e quasi-isome ic embeddings wi h uni o m cons an s.
1.7. HIERARCHICALLY HYPERBOLIC GROUPS 17
Mo eo e , he ollowing wo diag ams coa sely commu e (again wi h uni o m cons an s), o all
U, V PSsuch ha UĎVo U&V:
(1.2) Xφ//
πU

X1
πφ♦pUq

CUφ˚
U
//Cφ♦pUq
CUφ˚
U//
ρU
V

Cφ♦pUq
ρφ♦pUq
φ♦pVq

CVφ˚
V
//Cφ♦pVq
1.7 Hie a chically hype bolic g oups
De ini ion 1.7.1 (Hie a chically hype bolic g oup). We say ha a g oup Gis hie a chically
hype bolic i i ac s on a hie a chically hype bolic space pX,Sqsa is ying he ollowing condi ions:
1. The ac ion o Gon Xis p ope and cobounded;
2. Gac s co ini ely on S(i.e: wi h ini ely many o bi s), p ese ing he ela ions Ď,Kand &;
3. o each VPSand g, h PG, we ha e an isome y g:CVÑCgV such ha gh :CVÑCghV
is he composi ion o he isome ies gand h;
4. o all g1, g2PGwe ha e associa ed isome ies gi:CVÑCgiVsuch ha gπVpxq “ πgV pgxq
o e e y xPXand gρU
V“ρgU
gV whene e UĹVo U&V.
Rema k 1.7.2. By de ini ion, i pG, Sqis a hie a chically hype bolic g oup and gPG, mul ipli-
ca ion by gcoa sely sa is ies he wo diag ams o Equa ion (1.2). Howe e , i is always possible o
modi y he s uc u e o ob ain commu a i i y on he nose, as desc ibed in [36, Sec ion 2.1]. This
is he eason why he ou h i em in De ini ion 1.7.1 assumes equali y.
We end his sec ion wi h a ema k/wa ning:
Rema k 1.7.3. A hie a chically hype bolic space may admi se e al s uc u es. Conside he
ee g oup on wo gene a o s G“F2pa, bq. Since F2is hype bolic, pF2, F2uq is a hie a chically
hype bolic s uc u e. On he o he hand, Gspli s as xay ˚ xbyand he e o e F2is hype bolic
ela i e o xay,xbyu. Following he p e ious heo em we ob ain a non- i ial hie a chical hype bolic
s uc u e on F2.
To end he chap e , we include a ious no ions and ools exclusi e o hie a chically hype bolic
spaces ha a e a e needed o de elop he es o he hesis.
24 CHAPTER 1. PRELIMINARIES
( espec i ely o e e y WPSw), whe e pu:GÑGuis he canonical p ojec ion on he i s di ec
ac o , and πU:GuÑ2CUis he p ojec ion gi en in pGu,Suq.
I ollows ha o e e y UPSu he se πUpGwqis uni o mly bounded, and analogously o e e y
WPSw he se πWpGuqis uni o mly bounded. Mo eo e , he inclusions o he subg oups Guand
Gwin o Ga e ull, hie a chically quasicon ex hie omo phisms ha induce isome ies a he le el
o hype bolic spaces.
Example 1.9.2 (F ee p oduc o hie a chically hype bolic g oups). Le pGu,Suqand
pGw,Swqbe hie a chically hype bolic g oups. The ee p oduc Gu˚Gwis a hie a chically hype -
bolic g oup.
One way o seeing his is o ecall ha Gu˚Gwis hype bolic ela i e o Gu, Gwuand using
he ollowing heo em which shows ha g oups ha a e hype bolic ela i e o a collec ion o
hie a chically hype bolic subg oups a e hie a chically hype bolic. The p oo is al eady p esen ed
in [14, Theo em 9.1], bu we desc ibe he s uc u e he e o help wi h he exposi ion.
Theo em 1.9.3. [14, Theo em 9.1] Le Gbe a g oup ela i e o a ini e collec ion o pe iphe al
subg oups H1, . . . , Hku. I each Hican be endowed wi h a hie a chically hype bolic g oup s uc u e,
hen Gis a hie a chically hype bolic g oup.
P oo . Fo each i“1...,n and each le cose o Hiin G, ix a ep esen a i e gHi. Le gSibe
a copy o Siwi h i s associa ed hype bolic spaces and p ojec ions in such a way ha he e is
a hie omo phism HiÑgHiequi a ian wi h espec o he conjuga ion isomo phism HiÑHg
i.
Le p
Gbe he hype bolic space ob ained by coning-o Gwi h espec o he pe iphe als Hiu,
and le S“ p
GuYŮgPgŮiSgHi. The ela ion o nes ing, o hogonali y o ans e sali y be ween
hype bolic spaces belonging o he same copy SgHia e he same as in SHi. Fu he , i U, V belong
in wo di e en copies o di e en cose s, hen we impose ans e sali y be ween hem. Finally, o
e e y UPSgHiwe decla e ha Uis nes ed in o p
G.
The p ojec ions a e de ined as ollows: πp
G:GÑp
Gis he inclusion, which is coa sely su jec i e
and hence has quasicon ex image. Fo each UPSgHi, le ggHi:GÑgHibe he closes -poin
p ojec ion on o gHiand le πG
U“πHi
U˝ggHi, o ex end he domain o πU om gHi o G. Since
each πHi
Uwas coa sely Lipschi z on CUwi h quasicon ex image, and he closes -poin p ojec ion in
Gis uni o mly coa sely Lipschi z (Lemma 1.4.6), he p ojec ion πG
Uis uni o mly coa sely Lipschi z
and has quasicon ex image. Fo each U, V PSgHi, he a ious ρV
Uand ρU
Va e al eady de ined. I
UPSgHiand VPSg1Hj, hen ρU
V“πVpgg1HjpgHiqq. Finally, o U‰p
G, we de ine ρU
p
G o be he
cone-poin o e he unique gHiwi h UPSgHi, and ρp
G
U:p
GÑCUis de ined as ollows: o xPG,
le ρp
G
Upxq “ πG
Upxq. I xPp
Gis a cone poin o e g1Hj‰gHi, le ρp
G
Upxq “ ρSg1Hj
U, whe e Sg1Hjis
he Ď–maximal elemen o Sg1Hj. The cone-poin o e gHimay be sen anywhe e in CU.
By [14, Theo em 9.1], he cons uc ion abo e endows pG, Sqwi h a hie a chically hype bolic g oup

1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 25
s uc u e.
Rema k 1.9.4. In Theo em 4.2.2 we eadap his heo em o a mo e gene al s a emen .
A special ype o g oups ha can be buil induc i ely h ough di ec and ee p oduc s a e known
as g aph p oduc s o g oups:
De ini ion 1.9.5. [G aph p oduc s] Le Γ be a g aph and G“ G u PVpΓqbe a collec ion o
g oups. The g aph p oduc ΓGwi h espec o Gis de ined as
ΓG“ x˚ PVpΓqG | G , Gws “ 1ô , wu P EpΓqy
I Gis assumed o be a collec ion o δ-hype bolic g oups, hen by he p eceding discussion i
is na u al o expec ha he g aph p oduc ΓGis hie a chically hype bolic. This is indeed he
case, and we show a p oo o his in Theo em 3.3.7. E en mo e, we show ha g aph p oduc s
o hie a chically hype bolic g oups which ha e some e y na u al ex a p ope ies (in e sec ion
p ope y and clean con aine s) a e hie a chically hype bolic. We would also like o men ion ha
ha in [16], Be lyne and Russel gi e an independen p oo ha g aph p oduc s o hie a chically
hype bolic g oups a e hie a chically hype bolic ha imp o es Theo em 3.3.7 by emo ing he ex a
assump ions.
1.9.1 Hie a chically hype bolic s uc u es on g oups ac ing on ees
The main con ibu ions o his hesis is he in oduc ion o a wide a ie y o new examples o
hie a chically hype bolic g oups. These a e achie ed by es ablishing a combina ion heo em in his
class. I Cis a class o g oups, we usually e e o a esul as a combina ion heo em in Ci i
p o ides su icien condi ions ensu ing ha he undamen al g oup o a g aph o g oups in Cis again
in C. The Bes ina-Feighn combina ion heo em [17] o hype bolic g oups is such an example:
gi en a ini e g aph Go hype bolic g oups sa is ying ce ain condi ions, he esul ing undamen al
g oup is again hype bolic. Thei s a egy o p oo was o conside a me ic space (mo e p ecisely,
a ee o me ic spaces ob ained om he Bass-Se e ee o he g aph and he e ex/edge g oups
o G) and s udy he ac ion o he undamen al g oup on such space. This app oach u ned ou
o be e y success ul, and was la e applied in se e al o he ela ed con ex s. This is he case
o he combina ion heo em o [66] in he class o s ongly ela i ely hype bolic g oups, o o
he Hsu-Wise combina ion heo em in he con ex o g oups ac ing on cube complexes [51], o
Alibego i´c’s combina ion heo em o ela i ely hype bolic g oups [5]. On he o he hand, a mo e
dynamical app oach is unde aken by Dahmani [29] o ob ain ano he combina ion heo em o
ela i ely hype bolic g oups.
In Chap e 3 we p esen a combina ion heo em o hie a chically hype bolic g oups (Theo em 3.0.1).
As wi h he main de ini ion o hie a chical hype bolici y, unde s anding he ull s a emen o The-
o em 3.0.1 equi es he unde s anding o ce ain ools and echnicali ies. Chap e 3 and 2 a e
dedica ed o he de elopmen o said ools. We hus pos pone he ull o mula ion o he combi-
na ion heo em o Chap e 3.
26 CHAPTER 1. PRELIMINARIES
We now e iew Example 1.9.2 om a Bass-Se e heo y pe spec i e.
Example 1.9.6. Le G“Gu˚Gwbe he ee p oduc o wo hype bolic g oups. We desc ibe
he e an al e na i e hie a chically hype bolic s uc u e on G om ha in Example 1.9.2.
Le Guand Gwbe endowed wi h he i ial hie a chically hype bolic s uc u e. Recall ha each
e ex in he Bass-Se e ee Tassocia ed o Gu˚Gwco esponds o he se o cose s Po Gu
and Gwin Gu˚Gw. Le Xu, Xwdeno e he KpG, 1q-spaces associa ed o Guand Gw espec i ely
(i.e he CW-complexes such ha π1pXuq “ Guand π1pXwq “ Gw). Recall ha he join space
X“Xu_Xwis he KpG, 1q-space o Gu˚Gw.
π1(Xu) = Guπ1(Xw) = Gw
π1(Xu_Xw) = Gu∗Gw
The uni e sal co e
Xo Xcan be desc ibed as a space which has a combina o ial pa e n o
an in ini e ee. The ee is bipa i e wi h e ices labeled by he symbols Xuand Xw, ( i.e, he
Bass-Se e ee o G) as indica ed in Figu e 1.9.6. Mo eo e , he numbe o edges inciden on a
e ex labelled wi h Xua e in bijec ion wi h π1pXuqand likewise wi h Xwand π1pXwq. To each
e ex labeled wi h Xu( espec i ely Xw) we associa e he me ic space Ă
Xu( espec i ely Ą
Xw).
This desc ip ion o
Xcan be hough o as a ee o spaces:
De ini ion 1.9.7. [T ee o spaces] Le Tbe a simplicial ee and le V“VpTq, E “EpTq
deno e i s e ex and edge se espec i ely. A ee o spaces consis s o he quad uple
T“ pT, X u, Xeu, φe˘uq PV,ePE
whe e he maps φe˘:XeÑXe˘a e injec i e unc ions.
I Tis a ee o spaces, we de ine XpTq he o al space o Tas he me ic space whe e he
unde lying se is Ů PVX and adding edges o leng h one as ollows: i xPXe, we decla e φe´pxq
o be joined by an edge o φe`pxq. We de ine he dis ance on Xas ollows: i x, x1a e elemen s
on he same e ex space X , hen we say ha dXpx, x1q “ dX px, x1q. I x, x1a e joined by an
edge, we de ine dXpx, x1q “ 1. Gi en a sequence x1, . . . , xno poin s ei he joined by an edge o
li ing in he same e ex space, we de ine i s leng h o be řidXpxi, xi`1q. Fo gene al elemen s
1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 27
Xu
g
Xw
g
Xw
Xu
g
Xw
Figu e 1.2: Co e ing space o Xu_Xw
x, x1in X, we de ine he dis ance dXpx, x1qas he in imum be ween all leng hs o sequences such
ha x“x0, . . . , xk“x1.
I is no ha d o con ince onesel ha he o al space XpTqis quasi-isome ic o
X. Indeed, i we
collapse each pai o poin s in XpTqjoined by an edge o a poin we ob ain
X. This is clea ly
a quasi-isome y, as all ha we ha e done is collapse uni o mly bounded subspaces o XpTq o a
poin .
We now show ha Ghas a hie a chical hype bolic g oup s uc u e ob ained h ough he ac ion o
Gon
X. We begin by desc ibing a hie a chically hype bolic space s uc u e on
X.
(Index se ) I Tis he ee o spaces o
Xwe de ine he index se as S“ TuYŮ PV X u.
(Hype bolic spaces) We decla e ha CT“Tand ha CX “X o e e y e ex in T.
(P ojec ions) No e ha he e is a well-de ined map pT:XpTq Ñ Tob ained by collapsing each
e ex space o a poin . Mo eo e , i is s aigh o wa d o check ha his is a coa sely Lipschi z
map. We hen de ine he p ojec ion pT o be he p ojec ion πT om XpTq o T.
Fo each xPXpTqand each X we de ine he closes -poin p ojec ion p :XpTq Ñ X as ollows.
Le xPXbe an a bi a y elemen . I xPX , hen de ine p pxq:“x. I xRX , hen we de ine
p pxqinduc i ely. Le wbe he e ex such ha xPXw, suppose ha dTp , wq “ ně1, and
ha p p´q is de ined on all e ex spaces ha a e a dis ance s ic ly less han n om . Le
γbe he geodesic in Tconnec ing w o , le ebe i s i s edge, wi h e´“ . I ollows ha
dTpe`, q “ n´1. Then
p pxq:“p ´φe`˝φe´`pe´pxq˘¯,
whe e φe´is a quasi-in e se o φe´.
The a ious p ojec ions ρV
Ua e de ined as ollows: Fi s , i U, V co espond o e ex spaces Xu, X
espec i ely, hen ρU
Vis de ined as p pXuqand ρV
Uas pupX q. No e ha hese a e poin s, as he
edge spaces in Ta e i ial. I Uco esponds o Tand V o a e ex space X , we de ine ρV
Uas .
(Rela ions) Fo e e y pai o di e en e ices , w we impose ha X &Xwand ha X ĎT o
e e y PT.
28 CHAPTER 1. PRELIMINARIES
Wi h his s uc u e, all o he axioms o De ini ion 1.6.1 can be e i ied. We skip his e i ica ion
because he s uc u e is simple enough so ha all o he axioms a e ei he au oma ically sa is ied
o s aigh o wa d o p o e.
Rema k 1.9.8. Recall ha i we collapse e e y cose in P o a poin we ob ain he coned-o
Cayley g aph p
Gwi h espec o Gu, Gwu. Thus, i we de ine he map p
GÑTby sending a cose
o Guo Gw o i s co esponding e ex in T, hen we ha e a (coa sely)-well de ined map. I
ollows om [67, Lemma 3.1] ha his map yields a quasi-isome y be ween p
Gand T. Then, we
can simply swi ch he elemen Tin he s uc u e de ined abo e by p
G. By doing so, we ob ain he
same s uc u e desc ibed in Example 1.9.2 on G.
Le us now show ha he ac ion o Gon Xsa is ies he axioms o De ini ion 1.7.1. The i s
axiom is s aigh o wa d o check, as he quo ien X{Gis equal o X _Xw, which is a compac
space. The second one ollows om he ac ha e e y e ex space in Xis a copy o ei he X o
Xw, which means ha he e a e only ini ely many o bi s o elemen s in S. Fo he hi d axiom
conside gPGand X a e ex space. Then, g¨ “g is a e ex in T, and he associa ed
e ex space is Xg is an isome ic copy o X . To check he las i em, le gbe an elemen in
Gand le xPX. In pa icula , he e is some 1PTsuch ha xPX 1. I X is a e ex
space, hen he e is a unique pa h be ween and 1in T ha we call , 1s. Le p deno e he
closes -poin p ojec ion on o a e ex space desc ibed abo e. I eis he las e ex in , 1s hen
p pxq “ φe`p˚q, whe e Xe“ ˚. On he o he hand, i we apply g o , 1swe ob ain he pa h
g ,g 1sand he e o e pg pgxq “ φpgeq`p˚q “ gφe`p˚q “ gp pxq. One can a gue analogously o
ob ain ha gp pX 1q “ pg pXg 1qand, hus, gρU
V“ρgU
gV o e e y U, V in Ů PVX . I U“Tand
V“X , hen ρV
U“ and, he e o e, gρV
U“ρXg
T“ρgV
gU .
Rema k 1.9.9. The eade may ha e al eady no iced ha he g oup G“G ˚Gwwas known
o be hie a chically hype bolic om he beginning simply because he ee p oduc o hype bolic
g oups is hype bolic. This is indeed he case, bu we chose o desc ibe his speci ic s uc u e
because Theo em 3.0.1 gene alizes his idea. In ha sense, he example abo e is he mos basic
case possible o Theo em 3.0.1.
1.9.2 A cha ac e iza ion o hie a chical hype bolici y in hype bolic-2-
decomposable g oups
In he same way as he p esence o Z2as a subg oup o Gp e en s i om being hype bolic, he
p esence o he so-called unbalanced Baumslag–Soli a subg oups p e en s G om being hie a -
chically hype bolic. The ollowing ema k shows his ac :
Rema k 1.9.10. I Gis a hie a chically hype bolic g oup, hen Gcanno ha e a subg oup isomo -
phic o BSpn, mq “ xa, | an ´1“amy, wi h |n| ‰ |m|. Indeed, suppose he e is an embedding
1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 29
ι:BSpn, mqãÑG. We ha e ha ιpaqis an in ini e o de elemen o G. By [35, Theo em 7.1] and
[36, Theo em 3.1], ιpaqis undis o ed, which is a con adic ion.
This p omp s he ques ion: is he absence o unbalanced Baumslag–Soli a subg oups in a g oup
Genough o show ha Gis hie a chically hype bolic? This is indeed a e y big ques ion wi hou
assuming any hing on he g oup. Ins ead, in his hesis we p opose a mo e easonable one ha
assumes ha Gspli s o e i ually cyclic g oups. Mo e p ecisely, we conside g oups ha spli as
g aphs o g oups wi h 2-ended edge g oups. Fo he sake o b e i y, i Pis a p ope y o a g oup,
we say ha a g oup is P-2-decomposable i i spli s as a g aph o g oups wi h 2-ended edge g oups
and e ex g oups sa is ying p ope y P.
Conside ing g oups o his o m is no a no el y in geome ic g oup heo y. An impo an example
is he class o Z-2-decomposable g oups, also known as gene alized Baumslag–Soli a g oups (GBS
g oups). Al hough we will no di e deeply in he heo y o GBS g oups om a adi ional iewpoin ,
i is wo h no ing ha his class has been ex ensi ely s udied and shown o be an ex emely
ich objec o analyse om mul iple poin s o iew. To name a ew, GBS g oups ha e been
s udied in ela ion wi h JSJ decomposi ions ([39]), quasi-isome ies ([70]), au omo phisms ([58])
and cohomological dimension ([57]). Fo a gene al o e iew o esul s on GBS g oups we e e o
he su ey by Robinson ([72]).
One way o a oid unbalanced Baumslag–Soli a subg oup in Gis o impose a echnical condi ion
on Gcalled balancedeness. A g oup Gis said o be balanced i o e e y gPGo in ini e o de ,
whene e hgih´1“gj o some hPGi ollows ha |i| “ |j|. The no ion o balancedness
played an impo an ole in he heo y o g aphs o g oups. In [90], he au ho shows ha a
ee-2-decomposable g oup is subg oup sepa able i and only i i is balanced. In [78], he au ho s
ex end Wise’s esul o ( i ually- ee)-2-decomposable g oups, ob aining quasi-isome ical igidi y
o ce ain balanced g oups. In [28] he au ho s udies he ela ion be ween possible acylind ical
ac ions o ( o sion- ee)-2-decomposable g oups in connec ion wi h balancedness o such g oups.
A nai e conjec u e o make is ha a hype bolic-2-decomposable g oup Gis hie a chically hype -
bolic i and only i i is balanced. The las chap e o his hesis is dedica ed o p o e ha , up
o some issues wi h o sion on e ex g oups, he conjec u e holds (Theo em 4.2.15). In o de o
o mula e he esul s exp essly we in oduce he no ion o almos Baumslag–Soli a g oups:
De ini ion 1.9.11. Le Gbe a g oup. We say ha Gis an almos Baumslag–Soli a g oup i
i can be gene a ed by wo in ini e o de elemen s a, b PGsuch ha he equali y bamb´1“bn
holds o some n, m. In he pa icula case whe e |n|‰|m|we say ha Gis an unbalanced almos
Baumslag–Soli a g oup.
No e ha e e y almos Baumslag–Soli a g oup is he quo ien o some Baumslag–Soli a g oup.
Howe e , such quo ien map may no be an isomo phism.
We now ecall wo esul s due o Bes ina and Feighn ha ela e hype bolici y wi h almos
Baumslag–Soli a subg oups:

30 CHAPTER 1. PRELIMINARIES
Theo em 1.9.12 (Amalgams o e i ually cyclic g oups). Suppose ha G“G1˚CG2
is an amalgama ed ee p oduc whe e Giis hype bolic and Cis i ually cyclic. The ollowing
condi ions a e equi alen
1. Cis malno mal in ei he G1o G2;
2. Gis wo d hype bolic;
3. Gdoes no con ain BSp1,1q – Z2as a subg oup.
Theo em 1.9.13 (HNN ex ensions o e i ually cyclic g oups). Le Hbe a hype bolic
g oup and le Gbe he HNN ex ension G“ xH, yo e he i ually cyclic subg oups Aand B
whe e A ´1“B. Then he ollowing a e equi alen
1. Gis wo d hype bolic;
2. Gcon ains no almos Baumslag–Soli a subg oup;
3. o all hPH,|AXBh|ă 8 and ei he Ao Bis malno mal in H.
Using he almos Baumslag–Soli a g oup e minology, in Sec ion 4.1.4 we p esen he ollowing
gene aliza ion o he abo e heo ems o he class o hie a chically hype bolic g oups:
Theo em 1.9.14. Le Gbe a hype bolic-2-decomposable g oup. Then, Gis hie a chically hype -
bolic i and only i i con ains no unbalanced almos Baumslag–Soli a subg oups.
De ec ing almos Baumslag–Soli a subg oups: In gene al, checking whe he a gi en g aph
o g oups con ains an almos Baumslag–Soli a subg oup may be challenging. Fo his eason, we
in oduce he no ion o balanced edges. An edge eo a g aph o g oups Gis a balanced edge i o
e e y in ini e o de elemen gPGeand hPπ1pG´eq
i hgih´1“gj hen |i|“|j|.
We hen ha e he ollowing c i e ion o de ec almos Baumslag–Soli a subg oups.
Theo em 1.9.15. Le Gbe a g aph o g oups whe e none o he e ex g oups con ain dis o ed
cyclic subg oups. Then π1pGqcon ains a non-Euclidean almos Baumslag–Soli a subg oup i and
only i Ghas an unbalanced edge.
The p oo o Theo em 1.9.14 and 1.9.15 can be ound in Theo em 4.1.23.
1.9.3 A no e on o sion
The eade will ind ha Chap e 4 deals wi h hype bolic-2-decomposable g oups in wo sepa a e
se ings. Namely, when he g oup has o sion and when i does no .
1.9. CONSTRUCTING EXAMPLES OF HIERARCHICAL HYPERBOLICITY 31
I is pe haps wo h no ing ha he undamen al g oup o a g aph o i ually o sion- ee g oups
may no be i ually o sion ee (e en when he edge g oups a e cyclic), as he ollowing example
p o ided by A. Minasyan shows:2
Example 1.9.16. Le Hbe a g oup isomo phic o BSp2,3q“xa, b |ba2b´1“a3y. Since His
no esidually ini e, i s ini e esidual RespHq “ ŞKďH,|K:H|ă8 Kis non- i ial.
Le aPKbe non- i ial and le Gbe cons uc ed as G“ xH, b | b, Hs “ 1, b2“1y. No e ha
Gis i ually o sion ee. We hen cons uc he HNN ex ension Γ “ xG, | a ´1“aby. We
now show ha Γ is no i ually o sion- ee. Fi s , as HďΓ, we ha e ha RespHq ď RespΓq.
The e o e, RespΓqmus be non- i ial. Since RespΓqis a no mal subg oup o Γ, we ha e ha
a ´1“ab PRespΓq. We hus ob ain ha a´1pabq “ bPRespΓq. We conclude ha b, an elemen
o o de wo, belongs in e e y ini e index subg oup o Γ and he e o e Γ is no i ually o sion- ee.
Fo his eason, he main esul o Chap e 4 has wo o mula ions depending on he case:
Theo em 1.9.17. Le Gbe a hype bolic-2-decomposable g oup. The ollowing a e equi alen .
1. Gadmi s a hie a chically hype bolic g oup s uc u e.
2. Gdoes no con ain a dis o ed in ini e cyclic subg oup.
3. Gdoes no con ain a non-Euclidean almos Baumslag–Soli a g oup.
Mo eo e , i Gis i ually o sion- ee, condi ion (3) can be eplaced by
3’. Gdoes no con ain a non-Euclidean Baumslag–Soli a g oup.
A s aigh o wa d co olla y o his heo em is he ollowing.
Co olla y 1.9.18. Le G“H1˚CH2whe e Hia e hype bolic and Cis i ually cyclic. Then G
is a hie a chically hype bolic g oup.
As inal ema k, we belie e ha I em (3’) o Theo em 1.9.17 should be ue e en wi hou he
assump ion o Gbeing i ually o sion- ee. We e e he eade o he Ques ions sec ion in
Chap e 4 o u he discussion.
2h ps://ma ho e low.ne /ques ions/330632/is-an-hnn-ex ension-o -a- i ually- o sion- ee-g oup- i ually-
o sion- ee.
32 CHAPTER 1. PRELIMINARIES
Chap e 2
S uc u al esul s
The objec i e o his chap e is o ob ain s uc u al esul s ha will be necessa y o he de el-
opmen o Chap e 3. Mo eo e , his chap e in oduces a numbe o ools o analyze hie a chical
hype bolic spaces. The i s one is he in e sec ion p ope y (see De ini ion 2.1.1, and he discus-
sion a e he s a emen o Theo em 3.3.7), which in u n leads o he no ion o conc e eness. We
in oduce he la e no ion o exclude a i icial examples o hie a chically hype bolic spaces ha
ca y some undesi able ea u es. As we will see in his chap e , he in e sec ion p ope y has a
e y na u al de ini ion, and we conjec u e ha all hie a chically hype bolic spaces admi a hie -
a chically hype bolic s uc u e wi h he in e sec ion p ope y (see Ques ion 2.0.1 below). On he
o he hand, conc e eness is mo e echnical, bu ne e heless we p o e in P oposi ion 2.1.12 ha
any hie a chically hype bolic space wi h he in e sec ion p ope y can be supposed o be conc e e.
These p ope ies a e o independen in e es , and we expec hem o be o u he use.
Clean con aine s (see Rema k 1.6.2), a no ion in oduced o iginally by Abbo , Beh s ock, and
Du ham [2], is a echnical condi ion ha in he g aph o mul icu es se ing (see Subsec ion 1.5.2)
ansla es in o he ollowing: i VĎSis a subsu ace o he su ace S, hen Vand SzVa e disjoin ,
and any subsu ace disjoin om Vis con ained in o SzV. On he o he hand, he in e sec ion
p ope y is a condi ion ha we in oduce, and in he mapping class g oup se ing means ha ,
gi en wo subsu aces V, U ĎS, he subsu ace VXUis he bigges subsu ace o S ha is
con ained in bo h Vand U. The in e sec ion p ope y gi es o he index se S he s uc u e o a
la ice. A his poin , i is ins uc i e o no ice ha bo h VXUand SzVcould be non-connec ed
subsu aces o S, and indeed he hie a chically hype bolic s uc u e wi h clean con aine s and he
in e sec ion p ope y o he pan s decomposi ion g aph GpSqis ob ained conside ing all, possibly
non-connec ed, subsu aces o S.
We a e inclined o belie e ha any hie a chically hype bolic space admi s a hie a chically hype -
bolic s uc u e wi h he in e sec ion p ope y and clean con aine s:
Ques ion 2.0.1. Le pX, dXqbe a hie a chically hype bolic space. Does he e exis a hie a chically
hype bolic s uc u e Ssuch ha pX,Sqis a hie a chically hype bolic space wi h he in e sec ion
p ope y and clean con aine s?
33
40 CHAPTER 2. STRUCTURAL RESULTS
e e y UPS, and by De ini ion 1.8.1 he elemen zbelongs o FSεˆ eu. Le s0be he cons an
associa ed o he Dis ance Fo mula Theo em o he space pX,Sq, and conside sąmax ε, s0u.
The e exis K, C ą0 such ha
(2.4)
dpx, zq ď Kÿ
UPS dUpπUpxq, πUpzqquus`C
“K¨
˝ÿ
UPSSε
dUpπUpxq, πUpzqquus`ÿ
UPSzSSε
dUpπUpxq, πUpzqquus˛
‚`C
“Kÿ
UPSzSSε
dUpπUpxq, πUpzqquus`C.
No e ha dUpπUpxq, πUpzqq ď ε o e e y UPSzSε. Since sąε, om Equa ion (2.4) we conclude
ha dpx, zq ď C.
To comple e he p oo , no ice ha FSεˆ eucan be endowed wi h he hie a chical hype bolic
s uc u e SSε. Since X“NCpFSεˆ euq, he space pX,SSεqis hie a chically hype bolic, being
quasi isome ic o `FSεˆ eu,Sε˘, and i is conc e e by cons uc ion.
The in e sec ion p ope y in pX,SSεq ollows om he in e sec ion p ope y in pX,Sq.
Conc e eness will play an impo an ole in Lemma 2.3.2 and Theo em 2.3.3, a e he p oo o
Theo em 2.2.1.
Lemma 2.1.13. Gi en a ull hie omo phism φ:pX,Sq Ñ pX1,S1q, he e exis cons an s K, C ě0
and s, s1ą0such ha
ÿ
UPS dUpπUpxq, πUpyqquusďKÿ
U1Pφ♦pSq dU1pπU1pφpxq, πU1pφpyqqquus1`C@x, y PX.
P oo . Fo UPS, we deno e φ♦pUqby U1. As he hie omo phism is ull, he e exis s a uni o m
cons an ξsuch ha
(2.5) dU`πUpxq, πUpyq˘ďξdU1`πU1pφpxqq, πU1pφpyqq˘`ξ, @UPS,@x, y PX.
Choose sand s1such ha
s1:“s´ξ
ξą1.
Suppose ha sďdU`πUpxq, πUpyq˘ o a gi en UPS. Then, using Equa ion (2.5), we ob ain ha
(2.6) 1 ăs1ďdU1`πU1pφpxqq, πU1pφpyqq˘“ dU1`πU1pφpxqq, πU1pφpyqq˘uus1.
As sďdU`πUpxq, πUpyq˘we ha e ha dU`πUpxq, πUpyq˘uus“dU`πUpxq, πUpyq˘. I hen ollows

2.1. INTERSECTION PROPERTY AND CONCRETENESS 41
ha
dU`πUpxq, πUpyq˘uus“dU`πUpxq, πUpyq˘ďξdU1`πU1pφpxqq, πU1pφpyqq˘`ξ
ďξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξ.
(2.7)
The e o e, using Equa ion (2.6) and Equa ion (2.7), we ob ain
dU`πUpxq, πUpyq˘uusďξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξ
ďξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1`ξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1
“2ξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1.
(2.8)
On he o he hand, i sądU`πUpxq, πUpyq˘ hen
(2.9) dU`πUpxq, πUpyq˘uus“0ď2ξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1,
so he inequali y o Equa ion (2.8) is sa is ied also in his case.
Concluding, we use Equa ion (2.8) and Equa ion (2.9) o ob ain ha
ÿ
UPS dUpπUpxq, πUpyqquusďÿ
UPS
2ξ dU1`πU1pφpxqq, πU1pφpyqq˘uus1
“2ξÿ
UPS dU1`πU1pφpxqq, πU1pφpyqq˘uus1
“2ξÿ
U1Pφ♦pSq dU1`πU1pφpxqq, πU1pφpyqq˘uus1,
and he e o e he lemma is sa is ied wi h K“2ξand C“0.
Rema k 2.1.14. The a gumen o Lemma 2.1.13 can be used o show ha he e exis cons an s
¯
K, ¯
Cě0 and ¯s, ¯s1ą0 such ha
ÿ
U1Pφ♦pSq dU1`πU1pφpxqq, πU1pφpyqq˘uu¯sď¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`¯
C@x, y PX.
Lemma 2.1.15. Le φ:pX,Sq Ñ pX1,S1qbe a ull hie omo phism and Sbe he Ď-maximal
elemen in S. I S1“φ♦pSqand FS1ˆ euis a pa allel copy o FS1, hen πV1pFS1ˆ euq is coa sely
equal o πV1pφpXqq o all V1PS1
S1.
P oo . Le zPFS1and conside he uple ~
b“`πV1pzq˘V1PS1
S1. As zPFS1, he uple ~
bis κ-
consis en . The hie omo phism φis ull, he e o e S1
S1“φ♦pSqand
`πV1pzq˘V1PS1
S1“`πV1pzq˘V1Pφ♦pSq.
42 CHAPTER 2. STRUCTURAL RESULTS
As he ull hie omo phism φinduces uni o m quasi isome ies ¯
φ˚
V:CV1ÑCVa he le el o
hype bolic spaces, we ob ain a uple ~a “ paVqVPS, whe e aV:“¯
φ˚
V`πV1pzq˘ĎCV.
The uple ~a is κ1-consis en , and he e o e he e exis s xPX ha ealizes i , by [14, Theo em 3.1].
Exploi ing he ac ha he maps φ˚
V˝πVuni o mly coa sely coincide wi h he πV1˝φ(compa e
De ini ion 1.6.6) and in pa icula Equa ion (1.2)), we conclude ha he elemen φpxq ealizes he
uple ~
b:
(2.10) `πV1pzq˘V1Pφ♦pSq—`πV1pφpxqq˘V1Pφ♦pSq.
Tha is, he e exis s a cons an T1depending only on he ealiza ion Theo em [14, Theo em 3.1]
and he hie omo phism φsuch ha dV1pπV1pzq, πV1pφpxqqq ď T1 o e e y V1PS1
S1.
Con e sely, le φpxq P φpXqand conside he uple ~c:
cV1“$
’
’
’
’
&
’
’
’
’
%
πV1pφpxqq,@V1PS1
S1;
πV1peq,@V1PS1K
S1;
ρS1
V1@V1&S1o V1ĚS1.
Since ~c is a κ-consis en uple, he e exis s zPXsuch ha πVpzq — πVp~c q, and zbelongs o
FS1ˆ euby De ini ion 1.8.1. The e o e he e exis s T2such ha dV1pπV1pzq, πV1pφpxqqq ď T2 o
e e y V1PS1
S1.
P oposi ion 2.1.16. I φ:pX,Sq Ñ pX1,S1qis a ull hie omo phism be ween hie a chically hy-
pe bolic spaces, hen he spaces Xand FS1a e quasi isome ic, whe e S1is he image in S1o he
Ď-maximal elemen o S.
P oo . We de ine a map ψ:FS1ÑXand we p o e ha i is a quasi isome y. Le zPFS1, and
conside he uple ~
b“`πV1pzq˘V1PS1
S1. As zPFS1, he uple ~
bis κ–consis en . The hie omo phism
φis ull, so ha S1
S1“φ♦pSqand
`πV1pzq˘V1PS1
S1“`πV1pzq˘V1Pφ♦pSq.
As he ull hie omo phism φinduces uni o m quasi isome ies ¯
φ˚
V:CV1ÑCVa he le el o
hype bolic spaces, we ob ain a uple ~a “ paVqVPS, whe e aV:“¯
φ˚
V`πV1pzq˘ĎCV.
The uple ~a is κ1-consis en , and he e o e he e exis s xPX ha ealizes i by [14, Theo em
3.1]. Exploi ing he ac ha he maps φ˚
Vuni o mly coa sely commu e wi h he p ojec ions πV
(compa e De ini ion 1.6.6 and in pa icula Equa ion (1.2)), we conclude ha he elemen φpxq
2.2. PROOF OF THE MAIN THEOREM 43
ealizes he uple ~
b:
(2.11) `πV1pzq˘V1Pφ♦pSq—`πV1pφpxqq˘V1Pφ♦pSq.
De ine ψpzq:“x. The elemen xis no uniquely de e mined by he uple~
b, bu i is up o uni o mly
bounded e o .
Le us p o e ha ψis a quasi isome y. Indeed, le z1, z2PFS1. Using, in his o de , he Dis ance
Fo mula in X1, Rema k 2.1.14, and he ac ha φis a ull hie omo phism combined wi h he
Dis ance Fo mula in FS1, we ha e ha
(2.12)
dXpψpz1q, ψpz2qq ď Kÿ
UPS dUpπUpψpz1qq, πUpψpz2qqquus`C
ďK´K1ÿ
U1Pφ♦pSq dU1pπU1pz1q, πU1pz2qquu¯s`C1¯`C
ďK`K1pK2dX1pz1, z2q`C2q`C1˘`C.
On he o he hand, we ha e ha
(2.13)
dX1pz1, z2q ď K3ÿ
U1Pφ♦pSq dU1`πU1pz1q, πU1pz2q˘uus1`C3
ďK3´K4ÿ
UPS dU`πUpψpz1qq, πUpψpz2qq˘uu¯s1¯`C3
ďK3`K4pK5dXpψpz1q, ψpz2qq`C5q`C4˘`C3.
Equa ion (2.12) and Equa ion (2.13) p o e ha ψis a quasi-isome ic embedding.
Mo eo e , ψis coa sely su jec i e. Indeed, gi en an elemen xPX, he uple pπV1pφpxqqV1Pφ♦pSq
is consis en , and he e o e he e exis s a poin zPFS1coa sely ealizing i , ha is uni o mly close
o x.
We a e now eady o p o e he main esul o his chap e .
2.2 P oo o he main Theo em
Theo em 2.2.1. Le φ:pX,Sq Ñ pX1,S1qbe a ull hie omo phism wi h hie a chically quasicon ex
image, and le Sbe he Ď-maximal elemen o S. The ollowing a e equi alen :
1. φis coa sely lipschi z;
2. φis a quasi-isome ic embedding;
44 CHAPTER 2. STRUCTURAL RESULTS
3. he maps gφpXq:Fφ♦pSqÑφpXqand gFφ♦pSq:φpXq Ñ Fφ♦pSqa e quasi-in e ses o each
o he , and in pa icula quasi isome ies;
4. he subspace φpXq Ď X1, endowed wi h he subspace me ic, admi s a hie a chically hype bolic
s uc u e ob ained om he one o Xby composi ion wi h he map φ;
5. πWpφpXqq is uni o mly bounded o e e y WPS1zφ♦pSq.
P oo . The implica ions 3 ô5ñ1ô2ñ4ñ1 and 2 ñ3 a e enough o p o e he heo em.
5ñ1 By he Dis ance Fo mula applied in pX1,S1q, he e exis s s0such ha o e e y sąs0
he e exis s K1, C1ě0 o which
(2.14) dX1pφpxq, φpyqq ď K1ÿ
VPS1 dVpπVpφpxqq, πVpφpyqqquus`C1@x, y PX.
Also, he Dis ance Fo mula applied in pX,Sqimplies ha he e exis s s1such ha o e e y sąs1
he e exis K, C ě0 o which
(2.15) dXpx, yq ě K´1ÿ
UPS dUpπUpxq, πUpyqquus´C@x, y PX.
Now le x, y PX. By hypo hesis πWpφpXqq is uni o mly bounded o e e y WPS1zφ♦pSq. Le
Mbe his uni o m bound, and choose ssuch ha sąmax M, s0u. The e o e
ÿ
VPS1 dVpπVpφpxqq, πVpφpyqqquus“ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquus
and Equa ion (2.14) implies ha
dX1pφpxq, φpyqq ď K1ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquus`C1.
Using Rema k 2.1.14, we can choose ¯s, ¯s1ąs1and ¯
K, ¯
Cě0 o which
ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu¯sď¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`¯
C.
By aking ˜s“max s0,¯suwe ge
ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu˜sďÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu¯s
ď¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`¯
C.
2.2. PROOF OF THE MAIN THEOREM 45
As ¯s1ąs1, by he Dis ance Fo mula, Equa ion (2.14) and Equa ion (2.15) we ob ain
dX1pφpxq, φpyqq ď K1ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu˜s`C1
ďK1¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`K1¯
C`C1
ďK1¯
KpKdXpx, yq`KCq`K1¯
C`C1“RdXpx, yq`R1
o app op ia e cons an s Rand R1. The e o e, φis a coa sely lipschi z map.
1ô2 I φis a quasi-isome ic embedding, hen i is a coa sely lipschi z map.
Suppose now ha φis a coa sely lipschi z map. To conclude ha i is a quasi-isome ic embedding,
we need o p o e ha he e exis cons an s K, C ě0 such ha dXpx, yq ď KdX1pφpxq, φpyqq`C
o e e y x, y PX.
By he Dis ance Fo mula applied in pX,Sq, he e exis s s0so ha o e e y sěs0 he e exis
K1, C1ě0 so ha
dXpx, yq ď K1ÿ
UPS dUpπUpxq, πUpyqquus`C1,@x, y PX.
Also by he Dis ance Fo mula applied o pX1,S1q, he e exis s s1so ha o e e y sěs1 he e
exis K2, C2ě0 so ha
dX1pφpxq, φpyqq ě K´1
2ÿ
WPS1 dWpπWpφpxqq, πWpφpyqqquus´C2,@x, y PX.
By Lemma 2.1.13, we can choose ¯s, ¯s1ąs1and ¯
K, ¯
Cě0 such ha
ÿ
UPS dUpπUpxq, πUpyqquu¯sď¯
Kÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu¯s1`¯
C
ď¯
Kÿ
WPS1 dWpπWpφpxqq, πWpφpyqqquu¯s1`¯
C, @x, y PX.
Le s“max s0,¯su. Since sěs0and sě¯s, o any x, y PXwe ob ain ha
dXpx, yq ď K1ÿ
UPS dUpπUpxq, πUpyqquus`C1ďK1ÿ
UPS dUpπUpxq, πUpyqquu¯s`C1
ďK1˜¯
Kÿ
WPS1 dWpπWpφpxqq, πWpφpyqqquu¯s1`¯
C¸`C1
ďK1¯
K`K2dX1pφpxq, φpyqq` ¯
KC2˘`K1¯
C`C1“SdX1pφpxq, φpyqq`S1
o app op ia e cons an s Sand S1. The e o e, φis a quasi-isome ic embedding.

46 CHAPTER 2. STRUCTURAL RESULTS
2ñ4 I he map φis a quasi-isome ic embedding hen p4qis au oma ically sa is ied, because
hie a chical hype bolici y is p ese ed unde quasi isome ies (compa e wi h he ema k be o e
[12, Theo em G]).
4ñ1 As he hie omo phism is ull, e e y induced map φ˚
U:CUÑCpφ♦pUqq is a pξ, ξq-quasi
isome y, whe e ξis independen o UPS, ha is
ξ´1dUpπUpxq, πUpyqq´ξďdφ♦pUqpφ˚
UpπUpxqq, φ˚
UpπUpyqqq ď ξdUpπUpxq, πUpyqq`ξ
o all UPSand o all x, y PX.
By he Dis ance Fo mula applied in pX,Sq, he e exis s s0such ha o e e y sěs0 he e exis
K1, C1ě0 sa is ying
(2.16) dXpx, yq ě K´1
1ÿ
UPS dUpπUpxq, πUpyqquus´C1,@x, y PX.
We apply now he Dis ance Fo mula o he hie a chically hype bolic space pφpXq, φ♦pSqq. The e-
o e, he e exis s s1such ha o e e y sěs1 he e exis K2, C2ě0 sa is ying
(2.17) dX1pφpxq, φpyqq ď K2ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquus`C2,@x, y PX.
By Rema k 2.1.14, we can choose ¯s, ¯s1ąs0and ¯
K, ¯
Cě0 o which
(2.18) ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquu¯sď¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`¯
C, @x, y PX.
Fo s“max s1,¯su, combining Equa ion (2.16), Equa ion (2.17), and Equa ion (2.18), we ob ain
ha
dX1pφpxq, φpyqq ď K2ÿ
U1Pφ♦pSq dU1pπU1pφpxqq, πU1pφpyqqquus`C2
ďK2˜¯
Kÿ
UPS dUpπUpxq, πUpyqquu¯s1`¯
C¸`C2
ďK2¯
KpK1dXpx, yq`K1C1q`K2¯
C`C2“TdXpx, yq`T1
o app op ia e cons an s Tand T1. The e o e, φis a coa sely lipschi z map.
3ñ5By hypo hesis, gFS1:φpXq Ñ FS1and gφpXq:FS1ÑφpXqa e quasi in e ses o each o he ,
and by cons uc ion o ga e maps hey a e also coa sely lipschi z. The e o e FS1and φpXqa e
quasi-isome ic, whe e he quasi-isome y is gi en by gFS1, and in pa icula he e exis s Cą0
such ha
2.2. PROOF OF THE MAIN THEOREM 47
φpXq Ď NCpgφpXqpFS1qq.
Le WPS1zφ♦pSq. By he p e ious inclusion, he e exis s C1ą0, depending on Cand on πW,
such ha
(2.19) πWpφpXqq Ď NC1`πWpgφpXqpFS1qq˘.
Since he hie omo phism φis ull, φ♦pSq “ S1
S1. Mo eo e , by cons uc ion o ga e maps, he
se πWpgφpXqpFS1qq is uni o mly coa sely equal o pπWpφpXqqpπWpFS1qq, whe e pπWpφpXqq is he
closes -poin p ojec ion in CW o he quasicon ex subspace πWpφpXqq. Since WPS1zS1
S1, we
ha e ha diampπWpFS1qq ď αby [14, Cons uc ion 5.10] and, as a consequence, ha he e exis s
α1such ha diampπWpgφpXqpFS1qqq ď α1. The i s condi ion o he heo em ollows om his, and
Equa ion (2.19).
5ñ3We claim ha he e exis s Mą0 such ha
dX1pgFS1˝gφpXqpzq, zq ď M, dX1pgφpXq˝gFS1pyq, yq ď M, @zPFS1,@yPφpXq.
By applying he Dis ance Fo mula o he space pX1,S1q, he e exis s s0such ha o e e y sěs0
he e exis K1, C1ą0 such ha
dX1pgFS1˝gφpXqpzq, zq ď K1ÿ
U1PS1 dU1pπU1pgFS1˝gφpXqpzqq, πU1pzqquus`C1,@zPFS1.
By Lemma 2.1.7, diampπWpFS1qq ď ε o e e y WPS1zS1S1 o an app op ia e εą0. Fo
sěmax s0, εuand he p e ious equa ion, i ollows ha
(2.20) dX1pgFS1˝gφpXqpzq, zq ď K1ÿ
U1PS1
S1
dU1pπU1pgFS1˝gφpXqpzqq, πU1pzqquus`C1,@zPFS1.
Fo zPFS1, using he ac ha gFS1pzq “ z, we ob ain
(2.21)
dU1`πU1pgFS1˝gφpXqpzqq, πU1pzq˘“dU1pπU1pgFS1˝gφpXqpzqq, πU1pgFS1pzqqq ď
ďdU1pppπU1˝gφpXqpzqq, ppπU1pzqqq`2k
ďk1dU1pπU1pgφpXqpzq, πU1pzqq`c1`2k,
whe e p:CU1ÑπU1pFS1qis he closes -poin p ojec ion o he quasicon ex subspace πU1pFS1q Ď
CU1, and k1, c1deno e he mul iplica i e and addi i e cons an s associa ed o he coa sely lips-
chi z map p, and kdeno es he Hausdo dis ance be ween he (uni o mly) coa sely equal se s
48 CHAPTER 2. STRUCTURAL RESULTS
πWpgFS1pxqq and ppπWpxqq, o e e y xPX1.
By Lemma 2.1.15 he e exis s a cons an Tą0 such ha o e e y zPFS1 he e exis s φpxq P φpXq
o which dU1pπU1pφpxqq, πU1pzqq ď T o e e y U1PS1
S1. Since πU1pgφpXqpzqq coa sely equals
pπU1pφpXqqpπU1pzqq, we ob ain ha
dU1pπU1pgφpXqpzqq, πU1pzqq ď T1@U1PS1
S1.
By choosing an adequa e sin Equa ion (2.20), we conclude ha
dX1pgFS1˝gφpXqpzq, zq ď C1.
In o de o show ha dX1pgφpXq˝gFS1pyq, yqis uni o mly bounded o e e y yPφpXqle µą0
deno e he cons an such ha diampπWpφpXqqq ă µ o e e y WPS1zφ♦pSq “ S1zS1
S1. By he
Dis ance Fo mula he e exis s s0ą0 such ha o all sěs0 he e exis s K2, C2such ha
(2.22) dX1pgφpXq˝gFS1pyq, yq ď K2ÿ
U1PS1 dU1pπU1pgφpXq˝gFS1pyqq, πU1pyqquus`C2,@yPφpXq.
Since πU1˝gφpXq—pπU1pφpXqq ˝πU1, i ollows ha πU1pgφpXq˝gFS1q — pπU1pφpXqqpπU1˝gFS1q.
Mo eo e , i U1ĎS1, i ollows ha πU1˝gFS1—πU1, because πU1pFS1q — πU1pX1q o e e y
U1ĎS1. The e o e, we conclude ha πU1pgφpXq˝gFS1q — pπU1pφpXqq˝πU1. Fo any yPφpXqwe ha e
ha pπU1pφpXqq ˝πU1pyq “ πU1pyqand, he e o e, πU1pgφpXq˝gFS1pyqq — πU1pyq o e e y U1PS1
S1,
ha is o all U1PS1and o all yPφpXq, we ha e ha dU1pπU1pgφpXq˝gFS1pyqq, πU1pyqq ď ¯µ o
some cons an ¯µ.
Fo sěmax s0, µ, ¯µu, Equa ion (2.22) yields ha dpgφpXq˝gFS1pyq, yq ď C2, ha is he dis ance
is uni o mly bounded.
2ñ3 We claim ha pφpXq,S1S1qis a hie a chically hype bolic space. Since pX,Sqis a hie a chi-
cally hype bolic space and φpXqis quasi isome ic o X, we can endow φpXqwi h he hie a chically
hype bolic s uc u e gi en by he index se S. Fo VPS, he p ojec ions πV:φpXq Ñ CVin his
la e hie a chically hype bolic space a e de ined o be πV˝φ´1, whe e φ´1is a ixed quasi in e se
o φ:XÑφpXq, and πVa e he p ojec ions in he space pX,Sq.
Mo eo e , we can de ine he hie a chically hype bolic space pφpXq, φ♦pSqq. Fo V1Pφ♦pSq, ha
is o V1“φ♦pVqwi h VPS, he p ojec ions πV1:φpXq Ñ CV1a e de ined o be φ˚
V˝πV˝φ´1,
whe e φ´1and πVa e as be o e, and φ˚
V:CVÑCV1a e he (uni o m) quasi isome ies p o ided
by he hie a chically hype bolic space pX,Sq.
By De ini ion 1.6.6 we ha e ha φ˚
V˝πV—πV1˝φ, whe e πV1is he p ojec ion in he space
pX1,S1q. The e o e πV1—πV1˝φ˝φ´1, which uni o mly coa sely coincides wi h πV1, being φand
2.3. MAIN STRUCTURAL RESULTS 49
φ´1quasi in e ses o each o he . Thus pφpXq, φ♦pSqq is a hie a chically hype bolic space, whe e
we can ake he p ojec ions o be πV1 o all V1Pφ♦pSq, ins ead o πV1.
F om his poin , he a gumen o p o e ha he e exis s Mą0 such ha
dX1pgFS1˝gφpXqpzq, zq ď M, dX1pgφpXq˝gFS1pyq, yq ď M@zPFS1, y PφpXq
is exac ly he same as he one used in he p e ious implica ion 5 ñ3, and i is omi ed.
2.3 Main s uc u al esul s
Theo em 2.2.1 has se e al consequences. We s a wi h he ollowing:
Rema k 2.3.1. The combina ion heo em o Beh s ock, Hagen, and Sis o [14, Theo em 8.6] holds
wi hou he i s pa o hei ou h hypo hesis, ha is
i eis an edge o Tand Seis he Ď-maximal elemen o Se, hen o all VPSe˘, he
elemen s Vand φ♦
e˘pSeqa e no o hogonal in Se˘.
Indeed, his hypo hesis is used (compa e [14, De ini ion 8.23]) o de ine he uni o mly bounded
se s ρ Ws
Vswhen Wsand Vsa e ans e se equi alence classes whose suppo s do no in e sec .
By Theo em 2.2.1, ins ead o de ining
ρ Ws
Vs“cV˝ρφ♦
e`pSq
V
as done in [14, De ini ion 8.23], we can impose ha
ρ Ws
Vs“cV`πVe`pφe`pXeqq˘,
whe e eis he las edge in he geodesic connec ing T Ws o T Vs, wi h e`PT Vs, and cVis he
compa ison map om CVe` o he a o i e ep esen a i e o Vs. We will exploi his ac in he
p oo o Theo em 3.0.1 (compa e Subsec ion 3.2.3 and Equa ion (3.14)). The p oo o [14, Theo em
8.6], a e his modi ica ion, is no al e ed.
Lemma 2.3.2. Le φ:pX,Sq Ñ pX1,S1qbe a ull, coa sely lipschi z hie omo phism be ween hi-
e a chically hype bolic spaces such ha φpXqis hie a chically quasicon ex in X1, and le Sbe he
Ď-maximal elemen o S.
The e exis εand ε0such ha o all ε1ěε0, i pX,Sqis ε1-conc e e, wi h he in e sec ion p ope y
and clean con aine s, hen o any elemen WPS1we ha e ha WKsuppε`φpXq˘i and only i
WKφ♦pSq.
56 CHAPTER 3. A COMBINATION THEOREM
Assume i s ha uand a e e ices connec ed by a single edge esuch ha u“e´and “e`.
Then, he compa ison map is de ined as
c:“φ˚
e`˝φ˚
e´:CVuÑCV .
Whe e he maps φ˚
e`:CVeÑCVe`and φ˚
e´:CVeÑCVe´a e he quasi-isome ies induced by he
hie omo phisms φe`:XeÑXe`and φe´:XeÑXe´ espec i ely and φ˚
e´deno es a quasi in e se
o φ˚
e´.
Fo he gene al case, le γbe he geodesic in Tconnec ing u o , le uibe he i- h e ex o
his geodesic (so ha u“u0and “un o some na u al numbe ną0), and le eibe he
edge connec ing ui´1 o ui. Fo all i“1, . . . , n conside he hie omo phisms φe´
i:XeiÑXui´1
and φe`
i:XeiÑXui, and he induced quasi-isome ies φ˚
e´
i
:CVeiÑCVui´1and φ˚
e`
i
:CVeiÑCVui
om he hype bolic space associa ed o he ep esen a i e o Vsin Sei o he hype bolic spaces
associa ed o Vui´1and Vui espec i ely. Finally, le φ˚
e´
i
:CVui´1ÑCVeibe a quasi-in e se o he
map φ˚
e´
i
, o all i.
Then, he compa ison map cis de ined o be he composi ion o he p e ious quasi isome ies:
(3.2) c:“φ˚
e´
n˝φ˚
e´
n¨¨¨˝φ˚
e´
1˝φ˚
e´
1
:CVu0ÑCVun.
Rema k 3.1.5. I is a ac [14, Lemma 8.18] ha i he ca dinali y o suppo s is uni o mly
bounded, hen compa ison maps a e pξ, ξq-quasi-isome ies, o some uni o m (no depending on
he wo e ices uand ) cons an ξě1.
Rema k 3.1.6. I he edge hie omo phisms φe˘uePEo he ee o hie a chically hype bolic spaces
Tinduce isome ies a he le el o hype bolic spaces, hen we can choose in e se isome ies o he
maps φ˚
e˘. The e o e, om Equa ion (3.2) i ollows ha compa ison maps in his pa icula case
a e isome ies.
We eco d now he ollowing lemma, which is implici ly used in [14]. I s p oo ollows by applying
epea edly he (coa sely commu a i e) second diag am o Equa ion (1.2).
Lemma 3.1.7. Le Tbe a ee o hie a chically hype bolic spaces, and le Us, Vsbe wo equi -
alence classes such ha ei he Us& Vso UsĎ Vs. I compa ison maps a e uni o m quasi
isome ies, hen o all e ices u, PT UsXT Vs he se cpρUu
Vuqis coa sely equal o ρU
V , whe e
c:CVuÑCV is he compa ison map.

3.1. TREES OF HIERARCHICALLY HYPERBOLIC SPACES 57
3.1.1 T ees wi h deco a ions
Recall ha a ee o hie a chically hype bolic spaces (as de ined in De ini ion 3.1.1) is a uple
(3.3) T“´T, pX ,S qu PV, pXe,SequePE˘, φe˘:pXe,SeqÑpXe˘,Se˘qu¯,
whe e T“ pV, Eqis a ee, pX ,S qu PVuand pXe,SequePEua e amilies o uni o mly hie a -
chically hype bolic spaces, and φe`:pXe,Seq Ñ pXe`,Se`qand φe´:pXe,Seq Ñ pXe´,Se´qa e
hie omo phisms wi h cons an s all bounded uni o mly.
Recall ha , on Ů PVS one de ines he ollowing equi alence class: gi en an edge e“ , wu P E
and UPSe, impose φ♦
pUq o be equi alen o φ♦
wpUq, and ake he ansi i e closu e o his o
ob ain he desi ed equi alence ela ion. Gi en UPŮ PVS , i s equi alence class is deno ed by
Us.
In gene al, in a ee o hie a chically hype bolic spaces Ti migh happen ha wo dis inc equi a-
lence classes Us ‰ Vsa e suppo ed on exac ly he same e ices o he ee T, ha is T Us“T Vs.
This is no desi able, and in his subsec ion we desc ibe a sligh modi ica ion o he ee T(and
he e o e o he me ic space XpTqassocia ed o i ) ha ensu es ha Us “ Vsi and only i
T Us“T Vs. We achie e his by a aching o each e ex o Ta ee o uni o mly bounded
diame e , and e e o hese a ached ees as deco a ions. We deno e he ee ha is ob ained
wi h his p ocess by
T. As a consequence, he new suppo ees
T Uswill become la ge han he
o iginal ones (i.e. T UsĎ
T Us o each equi alence class Us).
All he hypo heses o Theo em 3.0.1 a e p ese ed by adding hese deco a ed ees ( u he mo e, he
me ic spaces associa ed o he wo ees o hie a chically hype bolic spaces a e quasi-isome ic),
and he e o e o he p oo o he heo em we will assume wi hou loss o gene ali y ha equi alence
classes a e disc imina ed by hei suppo s.
We now desc ibe how o deco a e he ee To hie a chically hype bolic spaces o Equa ion (3.3),
o ensu e ha Us“ Vsi and only i T Us“T Vs.
Fo any e ex PT, le S be he Ď-maximal elemen in S , le Ube any Ď-maximal elemen o
S z S uand le FUˆ ube a pa allel copy o he FUinside o X . Fo any such choice, we add
a new e ex ˜ and a new edge ˜econnec ing and ˜ . The me ic spaces X˜ and X˜ea e de ined o
be FUˆ u, wi h he induced me ic.
I ollows om [14, P oposi ion 5.11] ha `X ,SU˘and `X e,SU˘a e hie a chically hype bolic
spaces, o complexi y s ic ly lowe han `X ,S ˘. We e e o hese index se s as SU,
and SU,
e
espec i ely, whe e he exponen is added o keep ack o he choices o he Ď-maximal elemen
UPS z S u, and o he pa allel copy FUˆ u.
The hie omo phisms φ
e`and φ
e´a e de ined as ollows. A he le el o me ic spaces, φ
e`:X eÑ
X is he iden i y map and φ e´:X eÑX is he subspace inclusion. The map φ♦
e`:SU,
eÑSU,
is
he iden i y o he se SU, and φ♦
e´:SU,
eÑS is he inclusion. A he le el o hype bolic spaces,
he maps φ˚
e´,W , φ˚
e`,W :CWÑCWa e he iden i y o each WPSU,
e. I is s aigh o wa d
o check ha he commu a i e diag ams o De ini ion 1.6.6 a e sa is ied. Fu he mo e, since
φ♦
e`, φ♦
e´and φ˚
e`,W , φ˚
e´,W a e iden i y maps o inclusions, i ollows ha φ e`and φ e´a e ull
58 CHAPTER 3. A COMBINATION THEOREM
hie omo phism. Mo eo e , hey a e quasicon ex.
We epea his p ocess o any newly p oduced e ex, un il he complexi y o he esul ing hi-
e a chically hype bolic spaces is one. In pa icula , gi en a new e ex
wi h associa ed hie a -
chically hype bolic space `FUˆ u,SU˘no o complexi y one, conside a Ď-maximal elemen
VPSUz Uu. Conside mo eo e a pa allel copy FVˆ 1uo FVin FUˆ u, and epea he
p ocess o cons uc a new e ex wi h associa ed hie a chically hype bolic space `FVˆ 1u,SV˘.
We s ess ha FVis de ined in he hie a chically hype bolic space `FUˆ u,SU˘, and no in he
space `X ,S ˘ o which UPS .
We deno e by
T he ee o hie a chically hype bolic spaces ob ained om T ollowing his p ocess.
No ice ha XpTqcan be na u ally seen as a subspace o Xp
Tq, ha is XpTq Ď Xp
Tq. Mo eo e ,
as he complexi y o he hie a chically hype bolic spaces o Tis uni o mly bounded and each s ep
o he desc ibed p ocess educes he complexi y by one, he e exis s a uni o m cons an Csuch
ha NC`XpTq˘“Xp
Tq. In pa icula , he inclusion map ι:XpTqãÑXp
Tqis a quasi isome y,
and he e o e he wo spaces XpTqand Xp
Tqa e quasi isome ic.
In Xp
Tq, we deno e by „‹ he equi alence ela ion desc ibed in Subsec ion 3.1, by Us‹ he
equi alence class o UPŮ P
VS wi h espec o „‹, and by
T Us‹ he suppo o Us‹. No ice
ha
T Us‹XT“T Us o all UPŮ PVS , and ha o all
VPŮ P
VzVS he e exis s VPŮ PVS
such ha
V„‹V.
Rema k 3.1.8. In he con ex o hie a chically hype bolic g oups, deco a ing a ee Tamoun s
o he ollowing. Le be a e ex in Twi h associa ed g oup G, and conside he Bass-Se e ee
o G˚HH, whe e His a hie a chically quasicon ex subg oup o Go maximal, s ic ly smalle
complexi y, and he wo edge-embeddings a e gi en by he iden i y map idH:HÑHand by he
inclusion ι:HÑG. This Bass-Se e ee has one e ex 0wi h associa ed g oup G, and G:Hs
e ices iwhose associa ed g oups a e he G-cose s o he subg oup H, and edges eiconnec ing
0 o i.
In he ee T, we eplace he e ex by 0, and we add new e ices iand edges eiconnec ing
0 o i. To hese new e ices 0and i, we associa e he g oups gi en by he Bass-Se e ee o
he spli ing G˚HH.
Fo any new e ex iadded in such way, we epea he p ocess unless he e ex g oup Hhas
complexi y one.
Lemma 3.1.9. In he ee o hie a chically hype bolic spaces
Twe ha e ha Us‹“ Vs‹i and
only i
T Us‹“
T Vs‹.
P oo . One implica ion is i ial. Assume now ha
T Vs‹“
T Ws‹. I he complexi y o he wo
equi alence classes Vs‹and Us‹is di e en , hen he deco a ions added o he ee Ta e ees o
di e en diame e , and he e o e we canno ha e ha
T Vs‹“
T Ws‹. Thus, he equi alence classes
ha e he same complexi y, so nei he canno be p ope ly nes ed in o he o he .
By cons uc ion, in he ee
T he e a e e ices uand such ha Uand Va e Ď-maximal
3.1. TREES OF HIERARCHICALLY HYPERBOLIC SPACES 59
elemen s o S uand S , espec i ely. As
T Us‹“
T Vs‹, he equi alence class Us‹mus ha e a
ep esen a i e in S , and Vs‹mus ha e a ep esen a i e in S u. As nei he equi alence class can
be p ope ly nes ed in o he o he , i mus hen be ha Us‹“ Vs‹.
I he ee Tsa is ies he hypo heses o Theo em 3.0.1, hen also
Tdoes. We p o e his in he
ollowing lemmas.
Lemma 3.1.10. In he ee o hie a chically hype bolic spaces
T he edge hie omo phisms a e ull,
coa sely lipschi z, and hie a chically quasicon ex.
P oo . Le ebe an edge in
T. Two cases can occu : ei he eis an edge al eady in he ee T, o i
was added wi h he deco a ion o T.
I ewas al eady an edge in T, hen he edge hie omo phisms a e ull, coa sely lipschi z, and
hie a chically quasicon ex by he hypo heses o Theo em 3.0.1. On he o he hand, i eis a new
edge hen he wo maps φe´and φe`a e ull, hie a chically quasicon ex isome ic embeddings
(one is ac ually an isome y), by cons uc ion.
Lemma 3.1.11. The hie a chically hype bolic spaces o
Tha e he in e sec ion p ope y and clean
con aine s.
P oo . Le be a e ex o
T. I PT hen S has he in e sec ion p ope y and clean con aine s,
by he hypo heses o Theo em 3.0.1. I P
TzT, hen S
“SU,
coincides wi h SU, o some
UPŮ PVS . The e o e, S has in in e sec ion p ope y. Le PTbe he e ex such ha
UPS .
Suppose ha S “SU,
“SUdoes no ha e clean con aine s. The e o e, he e exis s WP
SUz Uusuch ha he se ZPSU|ZKWuis no emp y, and WMcon U
KW. By Lemma
2.1.5 we know ha con U
KW“U^con KW, whe e con KWis he o hogonal con aine o Win
S . Mo eo e WKcon KWby clean con aine s in S , and he e o e we each a con adic ion, as
con U
KWĎcon KU. Thus, SU,
has clean con aine s.
The a gumen o edge spaces is simila .
Lemma 3.1.12. Compa ison maps in
Ta e uni o mly quasi-isome ies.
P oo . Le , w be wo e ices in
Tand le Vs‹be an equi alence class suppo ed on bo h e ices,
wi h ep esen a i es V and Vw espec i ely. Conside he compa ison map c:CV ÑCVw, as
de ined in Equa ion (3.2). I bo h e ices al eady belong o TĎ
T, hen he map cis a uni o m
quasi-isome y by he hypo heses o Theo em 3.0.1.
I one e ex, say w, belongs in
TzT, and PT, conside he geodesic σin
Tconnec ing o
w. Le “ 0,... n“wbe he e ices o σ, such ha iis joined by an edge o i`i o all
i“0, . . . , n ´1. Then, he e exis s a maximal index i‹such ha i‹PTand i‹`1P
TzT; le
60 CHAPTER 3. A COMBINATION THEOREM
V‹be he ep esen a i e o Vsin S i‹. F om Equa ion (3.2) we see ha cis he composi ion o
c1:CV ÑCV i‹wi h c2:CV i‹ÑCVw. As no iced in he p e ious case, he map c1is a uni o m
quasi-isome y. Mo eo e , by cons uc ion, he map c2is an isome y, and he e o e cis a uni o m
quasi-isome y, being he composi ion o hese wo maps.
The las case o conside is when bo h e ices belong o
TzT. Depending on whe he he geodesic
σdoes no in e sec T, o does in e sec i , he map cwill be an isome y, o a composi ion o
h ee maps, wo o which isome ies and he emaining a uni o m quasi isome y.
The e o e, all compa ison maps a e uni o m quasi isome ies.
In iew o his, o he whole p oo o Theo em 3.0.1 we assume wi hou loss o gene ali y ha
equi alence classes a e di e en ia ed by hei suppo s al eady in he ee o hie a chically hype -
bolic space T, ha is Us“ Vsi and only i T Us“T Vs.
On he o he hand, o he p oo o Co olla y 3.3.1, ha is he applica ion o Theo em 3.0.1
o hie a chically hype bolic g oups, we will no deco a e he ee T. This is because, e en i a
hie a chically hype bolic g oup `G, S˘ac s on he index se S, he se o p oduc egions FUˆ
u | UPS, PEU(migh no be G-in a ian . The e o e, i migh happen ha he hie a chically
hype bolic space pXp
Tq,
Sq, whe e
Sdeno es he index se associa ed o he deco a ed ee
T,
does no admi a non- i ial ac ion o Gon o
S. We e e o Sec ion 3.3 o he comple e ea men
o his delica e poin .
We now de ine he hie a chically hype bolic s uc u e on his ee o hie a chically hype bolic
spaces.
3.2 Endowing a ee o HHS wi h an HHS s uc u e
As we ha e seen, whene e we a e p esen ed wi h a ee o me ic spaces T, i is possible o associa e
a me ic space XpTq o i called he o al space o T. Theo em 3.0.1 gi es su icien condi ions unde
which he o al space o a ee o hie a chically hype bolic spaces has a hie a chically hype bolic
space s uc u e. This sec ion is dedica ed o he p oo o Theo em 3.0.1 and is di ided in o h ee
subsec ions. In he i s sec ion we show how he index se is buil ; he second one desc ibes wha
he hype bolic spaces associa ed o each elemen in he index se a e. Finally, in he las subsec ion
we p o e Theo em 3.0.1 wi h he newly-de eloped elemen s.
3.2.1 Cons uc ion o index se
Rema k 3.2.1 (Conc e eness o he edge spaces). In he p oo o Theo em 3.0.1 we will
need o exploi conc e eness o he edge spaces, which is no an hypo hesis o he heo em. We
now explain why we can suppose, wi hou loss o gene ali y, ha all he hie a chically hype bolic
edge-spaces o Ta e ε-conc e e.
Le εě3 max α, ξuas in Lemma 2.1.7. I he edge spaces a e no all ε-conc e e, hen we apply
3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 61
P oposi ion 2.1.12 o each edge space Seo T o ob ain a sub-index se Se,ε ĎSesuch ha
pXe,Se,εqis ε-conc e e. No ice ha i Seis al eady ε-conc e e, hen Se,ε “Se.
Simila ly o wha is de ined in Subsec ion 3.1, de ine „ε o be he ansi i e closu e o „d,ε: o
any edge eand any UPSe,ε, we ha e ha φe`pUq „d,ε φe´pUq.
Doing so (and no de ining equi alence classes wi h espec o he equi alence class „o Subsec ion
3.1) will be c ucial o be able o apply Lemma 2.3.4 du ing he p oo o Theo em 3.0.1. Mo eo e ,
his does no a ec he hypo heses o he heo em, ha con inue o be sa is ied. Indeed, edge spaces
con inue o be uni o mly hie a chically quasicon ex in e ex spaces, wi h edge hie omo phisms
being ull and uni o mly coa sely lipschi z. Compa ison maps a e no a ec ed by his change (bu
he e migh be ewe o hem, as we a e conside ing possibly smalle edge-space index se s). Finally,
he in e sec ion p ope y is p ese ed by P oposi ion 2.1.12, and clean con aine s a e p ese ed by
Lemma 2.1.5.
In iew o Rema k 3.2.1, om now on we assume wi hou loss o gene ali y ha all edge spaces a e
ε-conc e e o some app op ia e ε, ha is ha he equi alence ela ions „εand „a e he same.
Le p
Tbe he esul o coning o he unde lying ee associa ed o he ee o spaces Twi h espec
o e e y suppo ee T Vs. We de ine he index se Sassocia ed o he ee o hie a chically
hype bolic spaces Tas
(3.4) S“S1 S2 p
Tu.
The se S1is
(3.5) S1:“´ğ
PV
S ¯{ „,
as de ined in Subsec ion 3.1.
Elemen s o S2co espond o suppo s o elemen s in S1:
(3.6) S2:“ T Vs| Vs P S1u.
We s ess ha all hese elemen s a e sub ees o he ee T, he ee a ached o he ee o
hie a chically hype bolic spaces T. By he ollowing lemma, he se S2is closed unde in e sec ions.
Lemma 3.2.2. Suppose ha T UsXT Vsis no emp y. Then he e exis s As P S1 o which
T As“T UsXT Vsand Us, VsĎ As.
P oo . Le V and U be he ep esen a i es o Vsand Usin he index se S , o all P
T UsXT Vs.

62 CHAPTER 3. A COMBINATION THEOREM
Fo all PT UsXT Vs, conside he se
Λ “ WPS |V , U ĎWu,
which is non-emp y since i con ains he maximal elemen o S .
Since V _W is, by de ini ion, he Ď-minimal elemen o S con aining bo h V and W , i is he
unique Ď-minimal elemen o Λ , which we deno e also by A . I T UsXT Vsconsis s o jus one
e ex , hen As“ V _U sis he desi ed equi alence class: as V sand U sa e nes ed in o
As, i ollows ha T AsĎT VsXT Us. The e o e T As“T VsXT Us.
I T VsXT Ushas mo e han one e ex, analogously o wha cons uc ed in he index se s o he
e ices, he e is a unique Ď-minimal elemen in he edge-index se Se ha we deno e by Ae, whe e
eis any edge ha con ains ep esen a i es o bo h Usand Vs.
Assume now ha , w PT UsXT Vsand ha he e is an edge e ha connec s hese wo e ices.
Then φ♦
pAeq “ A and φ♦
wpAeq “ Aw. The e o e
φ♦
pAeq “ φ♦
pVe_Ueq “ φ♦
pVeq_φ♦
pUeq “ V _U “A
by Lemma 2.1.3.
Thus A „Aw o all , w PT UsXT Vs, and we deno e by As he equi alence class o (any o
he) A s. By cons uc ion, Ashas a ep esen a i e whe e bo h Vsand Usha e, and hence
T UsXT VsĎT As.
On he o he hand we ha e ha Vsand Usa e nes ed in U _V s“ As, and he e o e T AsĎ
T UsXT Vsby Lemma 3.1.2. Thus, he lemma is p o ed.
Co olla y 3.2.3. Le Vs, Wsbe equi alence classes. Then, VsĎ Wsi and only i T WsĎT Vs.
P oo . I VsĎ Ws hen T WsĎT Vs, by Lemma 3.1.2. On he o he hand, i T WsĎT Vswe can
see ha T Ws“T WsXT Vs. By Lemma 3.2.2 he e exis s As P S1 o which T As“T WsXT Vs
and Vs, WsĎ As. I ollows ha T Ws“T As, and he e o e ha Ws“ As, because we a e
assuming ha he ee Tis deco a ed (compa e Lemma 3.1.9). Thus VsĎ Ws.
To de ine nes ing, o hogonali y, and ans e sali y, we p oceed as ollow. The elemen p
Tis he
Ď-maximal elemen .
Rela ions in S1a e as in [14]: wo „-equi alence classes Vsand Wsa e nes ed ( espec i ely
o hogonal), VsĎ Ws( espec i ely Vs K Ws), i he e exis a e ex PTand ep esen a i es
V , W PS such ha Vs“ V s, Ws“ W sand V ĎW ( espec i ely V KW ) in S . I
Vsand Wsa e no o hogonal and nei he is nes ed in o he o he , hen hey a e ans e se:
Vs& Ws.
3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 63
Rela ions in S2a e as ollows. Fo wo elemen s T Vs, T UsPS2, i T Vsis con ained as a se in
T Us hen T VsĎT Us, and ice e sa. O he wise hey a e ans e se, T Vs&T Us.
Rela ions be ween an equi alence class Wsand an elemen T VsPS2a e as ollows:
pc1qi WsĎ Vswe decla e Ws K T Vs;
pc2qi Ws K Vswe decla e WsĎT Vs;
pc3qo he wise, we decla e Ws&T Vs;
No ice ha Ws K T Vsi and only i T VsĎT Ws, by Co olla y 3.2.3.
3.2.2 Hype bolic spaces associa ed o he index se and p ojec ions
Le Cˆ
T“ˆ
T, which is p oduced om he ee Tby coning-o each sub ee T WsPS2.
Rema k 3.2.4. As soon as he e exis s a e ex space pX ,S qand wo o hogonal elemen s UKV
in S , hen he deco a ion ick o Subsec ion 3.1.1 implies ha all suppo s ees T WsPS2a e
p ope ly con ained in o he ee T. Indeed, i T Ws“T o some equi alence class, i mus hen be
ha T Usand T Vsa e p ope ly nes ed in o T Ws, and hus WsĎ Usand WsĎ Vsby Lemma
3.1.9. This con adic s he ac ha Us K Vs, and in pa icula ha he e is no equi alence class
nes ed in o bo h.
To each equi alence class Vswe associa e a a o i e e ex PT Vsand he a o i e ep esen a i e
V PS , so ha Vs“ V s. Then, de ine C Vs o be CV . By assump ion, he e exis s a uni o m
cons an ξě1 such ha o all e ices wsuch ha he e exis s WPSwwi h W„V , he
compa ison map c:V ÑWis a pξ, ξq-quasi-isome y.
Fo T WsPS2, le CT Ws:“p
T Wsbe he hype bolic space ob ained om he ee T Wsby coning-o
each sub ee T VsPS2p ope ly con ained in T Ws, ha is T VsĹT Ws.
De ine πp
T:XpTq Ñ p
Tas ollows: o xPX , de ine πp
Tpxq “ . No ice ha πp
Tis he composi ion
o he p ojec ion XÑTo Xon i s Bass-Se e ee wi h he inclusion o he ee Tin o p
T. Fo
all T WsPS2 he p ojec ion πT Wsis de ined analogously: o xPX , conside he closes -poin
p ojec ion o he e ex on o he sub ee T Wsin he ee T. The image o his poin unde he
inclusion map TãÑp
Tis πT Wspxq P CT Ws“p
T Ws. These p ojec ion maps πT Wsand he p ojec ion
map πp
Ta e uni o mly coa sely su jec i e, being su jec i e on he se o non-cone poin s.
Gi en Vs P Swi h a o i e ep esen a i e V˜ PS˜ , we de ine π Vs:XÑC Vsas ollows. I
πp
Tpxq “ is a e ex in he suppo o Vs, hen he e exis s a ep esen a i e V PS o he class
Vs, and π Vspxqis de ined o be
(3.7) π Vspxq:“c˝πV pxq Ď CV˜ “C Vs,
whe e c:CV ÑCV˜ is he compa ison map.
64 CHAPTER 3. A COMBINATION THEOREM
I πp
Tpxq “ is no in he suppo o Vs, le ebe he las edge in he geodesic connec ing o
T Vs, so ha e`PT Vs. De ine
(3.8) π Vspxq:“c˝πVe``φe`pXeq˘ĎCV˜ “C Vs,
whe e c:CVe`ÑCV˜ is he compa ison map.
Lemma 3.2.5. The p ojec ions de ined in Equa ion (3.7) and Equa ion (3.8) a e uni o m coa sely
lipschi z maps. Mo eo e , hey a e uni o mly coa sely su jec i e.
P oo . In Equa ion (3.7) he p ojec ions a e de ined as a composi ion o a uni o m quasi isome y
wi h a uni o m coa sely lipschi z map. The e o e, i su ices o show ha he p ojec ions in
Equa ion (3.8) a e uni o mly coa sely lipschi z oo.
To p o e so, no ice ha he edge econnec s he e ex e´, which lies ou side o T Vs, wi h he
e ex e`PT Vs, and no ice ha he e exis s a ep esen a i e Ve`PSe`o Vs. This means ha
Ve`U o any UPSe´, ha is Ve`PSe`zφ♦
e`pXeq.
As all hie omo phisms a e ull and coa sely lipschi z, in oking Theo em 2.2.1 we know ha he se
πVe`pφe`pXeqq a e uni o mly bounded. The e o e he p ojec ions as de ined in Equa ion (3.8) a e
uni o mly coa sely lipschi z, because he compa ison maps ca e uni o m quasi-isome ies and he
se s on which hey a e applied o a e uni o mly bounded.
These p ojec ions a e uni o mly coa sely su jec i e, because he p ojec ions o he e ex spaces
a e, ollowing he assump ion o Rema k 1.6.4.
3.2.3 P ojec ions be ween hype bolic spaces
Gi en an equi alence class Vs, de ine ρ Vs
p
T o be he suppo T Vso he equi alence class Vs,
which is uni o mly bounded in p
Tbecause i is coned-o . De ine ρp
T
Vs:p
TÑC Vsas ollows. Fo
wPTzT Vs, conside he geodesic connec ing w o T Vs, and le ebe i s las edge, so ha e`PT Vs.
De ine
(3.9) ρp
T
Vspwq:“c˝πVe``φe`pXeq˘ĎCV˜ “C Vs,
whe e c:CVe`ÑCV˜ is he compa ison map. I wPT Vs, hen ρp
T
Vspwqcan be chosen a bi a ily.
On he o he hand, i wPp
TzT, ha is wis a cone poin , hen de ine ρp
T
T Vspwq “ ρp
T
T Vspw1q, whe e
w1is an a bi a ily chosen e ex in he suppo ee associa ed o he cone-poin w.
Fo an elemen T WsPS2, de ine ρT Ws
p
T o be T Ws, and ρp
T
T Ws:p
TÑp
T Wsas ollows. Fo PT,
le ρp
T
T Wsp qbe he closes -poin p ojec ion (in he ee T) o on o T Ws. On he o he hand, i
Pp
TzT, ha is is a cone poin , hen de ine ρp
T
T Wsp q “ ρp
T
T Wsp 1q, whe e 1is any o he poin s
in he suppo ee associa ed o he cone-poin .
To de ine he p ojec ions ρ Vs
Wsbe ween („-classes o ) hype bolic spaces, we p oceed as ollows.
3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 65
I VsĎ Wso Vs& Ws, hen we de ine he p ojec ions as in [14, Theo em 8.6]. In pa icula ,
i VsĎ Ws he e exis e ices , w, 1such ha V , Wwa e he a o i e ep esen a i es o Vs
and Ws espec i ely, V 1and W 1a e ep esen a i es o Vsand Ws(possibly di e en om he
a o i e ones), and V 1ĎW 1. Mo eo e , le cV:CV 1ÑCV and cW:CW 1ÑCWwbe compa ison
maps. De ine
(3.10) ρ Vs
Ws“cW´ρV 1
W 1¯ĎCWw“C Ws,
which is a uni o mly bounded se in C Ws, and de ine ρ Ws
Vs:C Ws Ñ C Vsas
(3.11) ρ Ws
Vs“cV˝ρW 1
V 1˝¯
cW,
whe e ¯
cWis a quasi in e se o cWand ρW 1
V 1:CW 1ÑCV 1is he p ojec ion p o ided by he
hie a chical hype bolici y o he e ex space pX 1,S 1q.
Analogously, i Vs& Wsand he e exis s a e ex w1PTsuch ha Sw1con ains ep esen a i es
Vw1&Ww1o Vsand Ws, hen de ine
(3.12) ρ Vs
Ws“cW´ρVw1
Ww1¯ĎCWw“C Ws
and
(3.13) ρ Ws
Vs“cV´ρWw1
Vw1¯.
I he e is no common e ex o he suppo s o Vsand Ws, le , w be he closes pai o e ices
such ha S ,Swcon ain ep esen a i es V o Vsand Wwo Ws espec i ely, and le ebe he
las edge o he geodesic s a ing a wand ending a “e`. De ine
(3.14) ρ Ws
Vs“c˝πVe`pφe`pXeqq,
whe e c:CV ÑCV˜ is he compa ison map o he a o i e ep esen a i e. In a comple ely sym-
me ical way we also de ine ρ Vs
Ws.
Fo wo elemen s T Vsand T V1so S2, i T VsĹT V1s hen de ine ρT Vs
T V1s o be p
T Vs, which is
uni o mly bounded in p
T V1ssince i is coned-o . De ine ρT V1s
T Vs:p
T V1sÑp
T Vsas he closes -poin
p ojec ion.
I T Vs&T V1s, hen ρT V1s
T Vsand ρT Vs
T V1sa e ei he he closes -poin p ojec ions (i T Vsand T V1sdo
no in e sec ), o a e de ined o be p
T VsXp
T V1s, which by ( he p oo o ) Lemma 3.2.2 is equal o
p
T V _V1
s, whe e V and V1
a e ep esen a i es o Vsand V1sin a e ex PT VsXT V1s. No ice
ha i T VsXT V1sis no emp y, hen i is p ope ly con ained in bo h T Vsand T V1s, and he e o e
will be coned-o in bo h p
T Vsand p
T V1s.
Finally, we de ine p ojec ions be ween an equi alence class Wsand an elemen T VsPS2as
72 CHAPTER 3. A COMBINATION THEOREM
No ice ha
dWe`´πWe`pφe`pXeqq, ρS1
e
We`¯—dWe``πWe`pφe`pXeqq, πWe`pFS1
eq˘
ďKdpφe`pXeq,FS1
eq`K,
and so, by Theo em 2.3.3, we ha e ha
(3.21) dWe`´πWe`pφe`pXeqq, ρS1
e
We`¯ďKη `K.
Combining Equa ion (3.20) and Equa ion (3.21) we ob ain ha d Ws`ρ Vs
Ws, ρ Us
Ws˘is uni o mly
bounded.
Assume now ha T UsXT Ws“ H: in pa icula Us& Ws. By Lemma 3.1.2 we know ha
T VsĎT Us. The e o e, he e exis s an edge esepa a ing T Vs(and T Us) om T Ws, so ha
e`PT Ws.
As de ined in Equa ion (3.14), we ha e ha
ρ Vs
Ws“cW˝πWe`pφe`pXeqq “ ρ Us
Ws.
The e o e ρ Vs
Ws“ρ Us
Ws, and d Wspρ Us
Ws, ρ Vs
Wsq “ 0 is uni o mly bounded.
T W1s&T W2sLe T W1s, T W2sPS2sa is ying T W1s&T W2s, and le xPX. In his case, we always
ha e ha
min dT W1spπT W1spxq, ρT W2s
T W1sq, dT W2spπT W2spxq, ρT W1s
T W2sq(“0,
because ρT W1s
T W2sand ρT W2s
T W1sa e de ined as closes -poin p ojec ions i T W1sXT W2s“ H, o as he
(coned-o ) in e sec ion, i i is non-emp y.
T W1sĎT W2sLe T W1s, T W2sPS2sa is ying T W1sĎT W2s. Consis ency ollows, because o
all xPXwe ha e ha
πT W1spxq “ ρT W2s
T W1s`πT W2spxq˘.
The e o e diamCT W1s`πT W1spxqYρT W2s
T W1spπT W1spxqq˘“0, whe e CT Vs“p
T Vs, and he consis ency
inequali y is sa is ied.
Le T W3sPS2be such ha ei he
1. T W1sĎT W2sĹT W3s, o
2. T W2s&T W3s.
In ei he case we ha e ha ρT W1s
T W3sĎρT W2s
T W3s, and he e o e dT W3s`ρT W1s
T W3s, ρT W2s
T W1s˘“0.
Le now Vs P S1be such ha Vs&T W2sand Vs M T W1s. We wan o p o e ha d VspρT W1s
Vs, ρT W2s
Vsq
is uni o mly bounded. We now check e e y possible case. I he suppo o Vsdoes no in e sec
T W2s(and he e o e, does no in e sec T W1sĎT W2s), hen ρT W1s
Vs“ρT W2s
Vsand he claim is
sa is ied. I he suppo T Vsin e sec s bo h T W1sand T W2s, hen also in his case we ha e ha
ρT W1s
Vs“ρT W2s
Vs. Finally, i T Vsin e sec s T W2sbu no T W1s, hen ρT W1s
Vs“c˝πVe``φe`pXeq˘,

3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 73
whe e eis he las edge in he geodesic connec ing T W1s o T Vs, he e ex e`lies in T Vs, and
Ve`is he ep esen a i e o Vsin Se`. On he o he hand, ρT W2s
Vs“ρ W2s
Vs, and W2s& Vs. As
bo h classes Vsand W2sa e suppo ed on he e ex e`, we ha e ha ρ W2s
Vs“c˝ρW2e`
Ve`, whe e
W2e`is he ep esen a i e o W2sin ha e ex.
By Lemma 2.3.4 we ha e ha πVe``φe`pXeq˘is coa sely equal o ρ
Se`
Ve`, whe e
Se`“φ♦
e`pSeqand
Seis he Ď-maximal elemen o Se. The e o e d VspρT W1s
Vs, ρT W2s
Vsqis uni o mly bounded.
Vs&T WsLe T WsPS2. I T VsXT Ws“ H, hen
min d Vspπ Vspxq, ρT Ws
Vsq, dT WspπT Wspxq, ρ Vs
T Wsq(“0,@xPX.
Thus, suppose ha he in e sec ion is non-emp y. Since Vs&T Wsi ollows ha Vs& Ws. Sup-
pose ha dT Ws`πT Wspxq, ρ Vs
T Ws˘is big, so ha in pa icula xRT VsXT Ws“ρ Vs
T Wsand he
geodesic connec ing x o T Vspasses h ough he se T Ws.
By de ini ion, π Vspxq “ c˝πVe``φe`pXeq˘, and ρT Ws
Vs“ρ Ws
Vs“cpρWe`
Ve`q, whe e e`is he e ex
o he edge e ha belongs o T VsXT Ws, while e´PT WszT Vs, and Ve`and We`a e he
ep esen a i es o Vsand Ws espec i ely a he e ex e`.
Le Sebe he Ď-maximal elemen o Se. As he equi alence class Vsis no suppo ed in he
e ex e´, i ollows ha Ve`is no nes ed in o φ♦
e`pSeq “
Se. On he o he hand We`Ď
Se.
The e o e, ρWe`
Ve`and ρ
Se
Ve`coa sely coincide by De ini ion 1.6.1(4), and by Lemma 2.3.4 we ob ain
ha
πVe``φe`pXeq˘—ρ˜
Se
Ve`—ρWe`
Ve`,
ha is, π Vspxqand ρT Ws
Vscoa sely coincide. Thus, d Vs`π Vspxq, ρT Ws
Vs˘is uni o mly bounded.
VsĎT WsI he dis ance dT WspπT Wspxq, ρ Vs
T Wsq ą κ0, i ollows in pa icula ha πTpxq R
ρ Vs
T Ws“T VsXT Ws, and ha he geodesic in p
Tconnec ing x o ρ Vs
T Wspasses h ough he se
T WszT Vs. In his case, we ha e ha π Vspxq “ π Vs`πT Wspxq˘is equal o ρT Ws
Vs`πT Wspxq˘.
The e o e he consis ency inequali y is sa is ied also in his case.
(Fini e complexi y) I is enough o show ini e complexi y in S1and S2independen ly.
Fini e complexi y in S1 ollows om [14, Lemma 8.11]. Fo S2, no ice ha any chain o p ope
nes ings
T U1sĽT U2sĽ¨¨¨ ĽT Uns
induces he chain o p ope nes ings U1sĹ U2sĹ. . . Ĺ Unsin S1, by Co olla y 3.2.3.
As only equi alence classes a e allowed o be nes ed in o an in e sec ion o suppo s, and no ice
e sa, ini e complexi y is p o ed.
In pa icula , i ollows ha he complexi y o `XpTq,S˘is wice he complexi y o S1plus one,
and he complexi y o S1is max χ `1, whe e χ is he complexi y o he e ex space pX ,S q.
(La ge links) Le Ws P S1and x, x1PX. Suppose ha xPX and x1PX 1 o some , 1PT,
and le wbe he a o i e e ex o Ws. Le Edeno e he maximal o he cons an s E o he
Bounded Geodesic Axiom o he hie a chically hype bolic space pX ,S q.
74 CHAPTER 3. A COMBINATION THEOREM
Suppose ha , o some VsĎ Ws, we ha e d Vspπ Vspxq, π Vspx1qq ě E1, whe e E1depends on E
and on he quasi-isome y cons an s o he edge hie omo phisms. Then dVwpc˝πV pxq,c˝πV 1px1qq ě
E, o a ep esen a i e VwPSwo Vs. As he la ge links axiom holds in Sw, we ha e ha VwĎTi,
whe e TiPSwuN
i“1is a se o Nelemen s in Sw, whe e N“ d Wspπ Wspxq, π Wspx1qquand each
Tisa is ies TiĹWw. Mo eo e , he La ge Links Axiom in Swimplies ha d Wspπ Wspxq, ρ Tis
Wsq “
dWwpcW˝πW pxqq, ρTi
Wwq ď N o all i“1, . . . , N. Thus he la ge links axiom o elemen s Vs P S1
and Us P S Vs ollows.
We now conside he case o T WsPS2, and XPST Ws. This can happen bo h when Xis an
equi alence class, o when XPS2. We deal wi h he case XPS2in he ollowing lemma, whils
he case X“ Vs P S1is conside ed a e he lemma.
Lemma 3.2.7. Le x, x1PXand SPS2Y p
Tu. The se
Y“ XPS2|XĹS, dXpπXpxq, πXpx1qq ą 4u
is ini e. Mo eo e , he se o Ď-maximal elemen s in Yhas ca dinali y bounded linea ly in e ms
o he dis ance dS`πSpxq, πSpx1q˘.
P oo . Le σbe he geodesic in Tconnec ing “πTpxq o 1“πTpx1q. We begin by no icing
ha , i XXσ“ H, hen dXpπXpxq, πXpx1qq “ 0 because hese wo se s coincide, and he e o e
XRY. In pa icula , as nes ing be ween elemen s o p
TuYS2is inclusion, i σdoes no in e sec
S hen Ywill be emp y, and he lemma is i ially sa is ied.
Suppose now ha σin e sec s S, and conside he map ϕ:YÑPpσqde ined as ϕpXq “ XXσ,
whe e Ppσqis he se o subpa hs o σ. We i s p o e ha ϕis an injec i e map. Le X, X1PY
be such ha X‰X1and, looking o a con adic ion, suppose ha ϕpXq “ ϕpX1q, so ha
XXσ“X1Xσand he e o e XXσ“XXX1Xσ.
Since Xin e sec s σ, we ha e ha πXpxqand πXpx1qa e e ices o σ. The e o e πXpxqand πXpx1q
lie in XXσĂXXX1. Since XXX1is p ope ly con ained in bo h Xand X1, i will be coned-o
in bo h CXand CX1by cons uc ion. The e o e dXpπXpxq, πXpx1qq ď 2, which con adic s he
de ini ion o he se Y. The e o e he map ϕis injec i e, and he se Yis ini e.
We now claim ha , o elemen s X, X1PY, we ha e ha ϕpXq Ĺ ϕpX1qi and only i XĹX1.
Indeed, i XĹX1, ha is XĹX1, hen ϕpXq Ĺ ϕpX1q. On he o he hand, suppose ha
ϕpXq Ĺ ϕpX1q, and le X“T Vsand X1“T V1s, o some equi alence classes Vsand V1s. Since
ϕpXq “ XXσĹϕpX1q “ X1Xσ, we ha e ha
(3.22) XXσ“XXX1Xσ.
3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 75
Mo eo e , as XXX1“T VsXT V1s“T V_V1s, om Equa ion (3.22) we ob ain ha
(3.23) T VsXσ“T V_V1sXσ.
As VsĎ V_V1s, Lemma 3.1.2 implies ha T V_V1sĎT Vs. I T V_V1sis p ope ly nes ed in o T Vs,
hen T V_V1sis coned o in CT Vs“p
T Vs. Equa ion (3.23) implies ha dT VspπT Vspxq, πT Vspyqq “
2, which is a con adic ion since T VsPYby hypo hesis. The e o e, T V_V1s“T Vs, which implies
ha T VsĎT V1s, as desi ed.
We now show ha Ymax “ X1, . . . , Xnu Ď Y, he se o Ď-maximal elemen s in Y, has ca dinali y
a mos dSpπSpxq, πSpx1qq. Since e e y elemen o Ymax is p ope ly nes ed in o S, i ollows ha
i s suppo is coned o in CS“p
S. We now p o e ha XjXσĘ pXk1Y¨¨¨YXk qXσ o any
pai wise dis inc elemen s Xj, Xk1, . . . , Xk all belonging o Ymax.
The claim was jus p o ed o “1. Indeed, i XjXσĎXk1Xσ hen XjĎXk1, and his
con adic s he ac ha Xjand Xk1a e dis inc Ď-maximal elemen s o Y. Suppose ha XjXσĎ
pXk1YXk2qXσ, and le T Ujs, T Uk1sand T Uk2sdeno e Xj, Xk1and Xk2 espec i ely. In his case,
he e exis s a pa h in CXj om πXjpxq o πXjpx1q ha passes h ough he cone poin s o T Uj_Uk1s
and T Uj_Uk2s, which a e p ope ly nes ed in o Xj. Then, dXjpπXjpxq, πXjpx1qq ď 4, con adic ing
he assump ion ha XjPYmax.
On he o he hand, assume ha XjXσĎ pXk1YXk2Y. . . YXk qXσwhe e ą2, ki‰j o all
i,ka‰kb o all a‰b, and he e does no exis ki‰kjsuch ha XjXσĎ pXkiYXkjqXσ. We
claim ha he e exis s ssuch ha XksXσĎXjXσ.
Indeed, assume wi hou loss o gene ali y ha he endpoin s o XjXσa e con ained in Xk1Xσ
and Xk Xσ espec i ely. By hypo hesis, XjXσcanno be en i ely con ained in pXk1YXk qXσ.
The e o e, he e exis s PXjXσzpXk1YXk qXσ, ha is PXksXσ o 1 ăsă . No e ha
XksXσcanno con ain ei he o he endpoin s o XjXσ, since ha would imply ha XjXσ
is con ained in ei he pXk1YXksq X σo pXk YXksq X σ. As a consequence we ob ain ha
XksXσĎXjXσ, which is a con adic ion, since Xksis maximal wi h espec o nes ing.
F om he e we can conclude ha |Ymax|ďdSpπSpxq, πSpx1qq. Indeed, gi en any Ď-maximal elemen
XiPYmax and i s cone poin i, he ollowing dicho omy holds: ei he iis a e ex in he geodesic
pa h pσ, o no , whe e pσis a geodesic pa h in CSconnec ing πSpxq o πSpx1q. In he la e case,
i mus be ha pσcon ains ei he one o wo edges o he suppo Xi. The e o e, he bound is
p o ed.
The e o e, i dT UspπT Uspxq, πT Uspx1qq ą 4 o some T UsPSSz Su, ha is T UsPY, hen T UsĎX
o some Ď-maximal elemen Xo he se Y.
We now add ess he case when Xis an equi alence classes X“ Vs P ST Ws. By de ini ion,
VsĎT Wsi and only i Vsis o hogonal o Ws. In pa icula , i ollows ha T VsXT Ws‰ H.
76 CHAPTER 3. A COMBINATION THEOREM
I T Vsdoes no in e sec he geodesic σ hen he dis ance d Vs`π Vspxq, π Vspx1q˘is equal o ze o
by Equa ion (3.8), because he edge eappea ing in he ci ed equa ion will be he same o bo h x
and x1.
Now assume ha T VsXσ‰ H. As a is sub-case, suppose ha σXT Wsis emp y, le
(3.24) I:“ VsĎT Ws|T VsXσ‰ H(,
and no ice ha Icould be in ini e. Conside he geodesic αconnec ing T Ws o σin he ee T,
and no ice ha αhas a leas one edge, being T Wsand σdisjoin . Fo Vs P I, we ha e ha T Vs
in e sec s bo h T Wsand σ, and he e o e αis con ained in T Vs, being Tis a ee. Thus he se
T WsXŞ VsPIT Vsis no emp y, because (a leas ) he ini ial e ex on he geodesic αbelongs o
his in e sec ion.
Le he se Iindex I, ha is I“ VisuiPI. Wi hou loss o gene ali y, we can suppose ha each Vi
is he ep esen a i e o Visin he e ex space pX ,S q. Le S PS be he Ď-maximal elemen ,
and no ice ha VisĎ S s o all iPI. Fu he mo e, no e ha VsĎ ŽiPIVis o all Vs P Iand
le V_sdeno e ŽiPIVis. The e o e, in his i s sub-case, La ge Links is sa is ied by he amily
YY V_su o he elemen s T WsPSand x, x1PX.
Fo he second sub-case, suppose ha σXT Wsis no emp y, and le 1, . . . , nube he ini ely
many e ices o σXT Ws( he e can be only ini ely many such e ices because σis a geodesic).
Analogously o Equa ion (3.24), o all iPσXT Wsde ine
I i“ VsĎT Ws| iPT VsXσ(,
and no ice ha I“ŤI i. As in he p e ious case, o each I iconside S is, and no ice ha
VsĎ S is o all Vs P I i, o all i“1, . . . , n. The e o e, La ge Links o an elemen T WsPS2
is sa is ied conside ing he se YY V 1
_s,..., V n
_su.
No ice ha , in bo h sub-cases, we bounded he ca dinali y o he se s YY V_su and YY
V 1
_s,..., V n
_su in e ms o σ, ha is in e ms o dTpx, x1q. As dT WspπT Wspxq, πT Wspx1qq is
bounded om abo e by dTpx, x1q, we ob ained he desi ed bound on he ca dinali y o hese se s.
Combining hese bounds wi h Lemma 3.2.7, we conclude he p oo o La ge Links o he case
XĹT Ws.
Finally, we p o e La ge Links o he Ď-maximal elemen p
T. F om Lemma 3.2.7 applied wi h
S“p
T, he e a e only ini ely many (and he numbe depends only on he dis ance in p
T om x o
x1) elemen s XPS2such ha dXpπXpxq, πXpx1qq is big. On he o he hand, o an equi alence
class VsĎp
T, he dis ance d Vspπ Vspxq, π Vspx1qq can be big only i he suppo T Vsin e sec s
he geodesic σconnec ing o 1(o he wise, i would be ze o). Le S1, . . . , Snbe he Ď-maximal
elemen s o all he ini ely many edges in σXT Vs. We ha e ha VsĎ Sis o all i“1, . . . , n.
The e o e, he se YY S1, . . . , Snuis he se o signi ican elemen s o he Axiom.
Le E1be he cons an ha sa is ies he La ge Links Axiom o he (uni o mly) hie a chically
hype bolic e ex spaces (see De ini ion 1.6.1), and le Eąmax 2, E1u. Then La ge Links is
sa is ied wi h his cons an E.
3.2. ENDOWING A TREE OF HHS WITH AN HHS STRUCTURE 77
(Bounded geodesic image) Conside WsĹp
T, and le γbe a geodesic in ˆ
T. I γXT Vs“
H, le ebe he las edge in he geodesic connec ing γ o T Vs, and suppose e`PT Vs. Then
ρˆ
T
Vspγq “ cW˝πVe`pφe`pXeqq is a uni o mly bounded se . I no , hen γin e sec s ρ Vs
p
T. The cases
VsĎT W1s,T W1sĎT W2s, and T W1sĎˆ
T, whe e T W1s, T W2sPS2, a e analogous.
Le Ws P S, le VsĎ Ws, and le γbe a geodesic in C Ws “ CWw(whe e wis he a o i e
e ex o Wsand WwPSwis he a o i e ep esen a i e). Le Vwbe he ep esen a i e o Vs
suppo ed in he e ex w, so ha ρ Vs
Ws“ρVw
Ww. The Bounded Geodesic Image Axiom in his
case ollows because i holds in he e ex space pXw,Swq(no ice ha he cons an Echanges
acco ding o he quasi-isome y cons an o he compa ison maps).
(Pa ial ealiza ion) No ice ha wo elemen s T W1sand T W2so S2a e ne e o hogonal.
Conside k`1 pai wise o hogonal elemen s V1s,..., Vks, T WsPS, and le piPπ VispXq Ď C Vis,
o i“1, . . . , k, and SPp
T Ws.
By de ini ion o o hogonali y, T VisXT Vjs‰ H o all i‰j,T WsĎT Vis o all i“1, . . . , n,
and in pa icula T WsĎŞk
i“1T Vis. Conside a e ex PT Ws ha is no a cone poin and
has dis ance a mos one om S, ha is PTXT Wsand dT Wsp , Sq ď 1. As PT Vis
o all i“1, . . . , k, wi hou loss o gene ali y we can suppose ha Viis an elemen o S , by
choosing ep esen a i es. We ha e ha ViKVj o all i‰j. Compa ison maps a e uni o m quasi-
isome ies, and piPπ VispXq, he e o e he elemen cippiqis uni o mly close o he se πVipXq o
all i“1, . . . , k, whe e ci:C Vis Ñ CViis he compa ison map. Fo i“1, . . . , k, le p
iPπVipXqbe
a poin such ha dVi`p
i,cippiq˘is uni o mly bounded.
By Pa ial ealiza ion in he e ex space pX ,S q, he e exis s xPX such ha dVipπVipxq, p
iq
is uni o mly bounded o all i. As compa ison maps a e uni o m quasi-isome ies, we ob ain ha
d Vispπ Vispxq, piqis uni o mly bounded o all i. Mo eo e , dT WspπT Wspxq, Sq “ dT Wsp , Sq ď 1.
I VisĎ Us, hen Ushas a ep esen a i e U PS such ha ViĎU . The e o e d Uspπ Uspxq, ρ Vis
Usq
is uni o mly bounded, because xis a ealiza ion poin o Viuk
i“1, and compa ison maps a e uni o m
quasi isome ies.
I VisĎT Us, hen ρ Vis
T Us“T VisXT Usand πT Uspxq P ρ Vis
T Us. The e o e, dT Us`πT Uspxq, ρ Vis
T Us˘“
0. Analogously, o T WsĎT Uswe ha e ha dT Us`πT Uspxq, ρT Ws
T Us˘“0. This a gumen also
applies when conside ing he Ď-maximal elemen , he e o e p o ing ha dp
T`πp
Tpxq, ρT Ws
p
T˘“0 and
dp
T`πp
Tpxq, ρ Vis
p
T˘“0.
Le now Vis& Us. Ei he T UsXT Vis“ H, in which case he dis ance d Uspπ Uspxq, ρ Vis
Usqis
uni o mly bounded, o T UsXT Vis‰ H, in which case Ushas a ep esen a i e U PS ha is
ans e se o Vi. The e o e, in he la e case he dis ance d Uspπ Uspxq, ρ Vis
Usqis again uni o mly
bounded, because i is in he e ex space X , and compa ison maps a e uni o m quasi-isome ies.
I Vis&T Us hen πT Uspxq P ρ Vis
T Us, and he e o e dT Us`πT Uspxq, ρ Vis
T Us˘“0. Fo he las case,
suppose ha T Ws& Us o some Us P S1. I he suppo o Usdoes no in e sec T Ws, hen
π Uspxq P ρT Ws
Us. So, suppose ha T Wsin e sec s T Us. Again using Lemma 2.3.4, we can conclude.
I T Ws&T Usand T WsXT Us‰ H, hen he sub ee T WsXT Us“T W_Usis s ic ly con ained
in T Us. The e o e, T WsXT Usis coned-o in CT Us“p
T Us. Since πT Uspxq P T WsXT Us, we
ob ain ha dT UspπT Uspxq, ρT Ws
T Usq ď 2. On he o he hand, i T WsXT Us“ H hen πT Uspxq “

78 CHAPTER 3. A COMBINATION THEOREM
ρT Ws
T Us“e`PT Us, whe e eis he las edge in he geodesic sepa a ing T Ws om T Us, and he e o e
dT UspπT Uspxq, ρT Ws
T Usq “ 0. By de ini ion, no elemen o S1can be nes ed in o an elemen o S2.
The e o e, all he ele an cases ha e been conside ed.
(Uniqueness) Suppose x, y PXa e such ha dRpπRpxq, πRpyqq ď K, o all RPS. In pa ic-
ula , we ha e ha dp
T`πp
Tpxq, π p
Tpyq˘ďK, ha dS`πSpxq, πSpyq˘ďK o all SPS2, and ha
d Vs`π Vspxq, π Vspyq˘ďK o all Vs P S1.
Suppose ha he dis ance in p
T om πp
Tpxq o πp
Tpyqis ealized by a pa h only consis ing o e ices
o TĎp
T, and le
0“πTpxq, 1, . . . , k´1, πTpyq “ k,
be hese e ices, whe e kďK. In pa icula , no ou consecu i e e ices can belong o he same
suppo ee, because his would p oduced a sho e pa h in p
Tjoining x o y.
We ha e ha dXpx, yq ď řk
i“0dX i`g ipxq,g ipyq˘`k. Mo eo e , o all i“0, . . . , k we ha e
ha he dis ance dX i`g ipxq,g ipyq˘is uni o mly bounded. Indeed, i his is no he case, by
Uniqueness in he hie a chically hype bolic space pX i,S iq, he e exis s VPS isuch ha
dV`πVpg ipxqq, πVpg ipyqq˘is no bounded. By [14, Lemma 8.18] and Theo em 2.2.1, we ha e
ha dV`πVpg ipxqq, πVpg ipyqq˘and d Vs`π Vspxq, π Vspyq˘coa sely coincide, and he e o e he
la e is no bounded. This con adic s he ac ha d Vs`π Vspxq, π Vspyq˘ďK, and hus
dX i`g ipxq,g ipyq˘ďζ“ζpKqis uni o mly bounded, as claimed. The e o e, dXpx, yq ď ζ1pKq,
o some uni o m bound ζ1pKq.
Suppose now ha in he geodesic σin ˆ
Tconnec ing πˆ
Tpxq o πˆ
Tpyq he e is a cone poin . The e-
o e, he e exis s an elemen T W1sPS2con aining wo poin s x1and y1in his geodesic ( ha ,
he e o e, ha e dis ance wo in ˆ
Tsince T Wsis coned-o in p
T). As T W1sPS2, we ha e ha
dT W1s`πT W1spx1q, πT W1spy1q˘“dT W1spx1, y1q ď K. Ei he he geodesic σ1in CT W1s“p
T W1s
connec ing hese wo poin s only consis s o e ices o T, o he e a e cone poin s, and he e o e
an elemen T W2sPS2con aining wo elemen s x2, y2o he geodesic σ1.
As complexi y in S2is ini e and nes ing coincides wi h inclusion, his p ocess mus end a e a
ini e numbe o s eps ( ha depends only on K). The e o e, he e exis s a geodesic in Tconnec ing
πˆ
Tpxq o πˆ
Tpyq, whose leng h is bounded om abo e by a unc ion in K. Repea ing he a gumen
gi en be o e, we conclude ha dXpx, yqis uni o mly bounded.
This concludes he p oo o hie a chical hype bolici y o he space `XpTq,S˘.
3.3 Applica ions
Theo em 3.0.1 has wo main applica ions. The i s one is a combina ion heo em on hie a chi-
cally hype bolic g oups (Co olla y 3.3.1). The second one is o g aph p oduc s o hie a chically
hype bolic g oups (Theo em 3.3.7). We now show hei p oo s.
3.3. APPLICATIONS 79
3.3.1 G aph o hie a chically hype bolic g oups
Co olla y 3.3.1. Le G“`Γ, G u PV, GeuePE, φe˘:GeÑGe˘uePE˘be a ini e g aph o hie -
a chically hype bolic g oups. Suppose ha :
1. each edge-hie omo phism is hie a chically quasicon ex, uni o mly coa sely lipschi z and ull;
2. compa ison maps a e isome ies;
3. he hie a chically hype bolic spaces o Gha e he in e sec ion p ope y and clean con aine s.
Then he g oup associa ed o Gis i sel a hie a chically hype bolic g oup.
We begin wi h he ollowing lemma, in which we use he no a ion o Sec ion 3.1.1.
Lemma 3.3.2. Le Tbe a ee o hie a chically hype bolic spaces and
Tbe he co esponding
deco a ed ee. Then
1. o e e y suppo ee T VsPS2π
T Vs‹pXpTqq is isome ic o CT Vs, and quasi-isome ic o
C
T Vs‹, o all suppo ees ;
2. π Vs‹pXpTqq is isome ic o π VspXpTqq, and quasi-isome ic o π Vs‹pXp
Tqq, o all equi -
alence classes Vs P S1;
3. XpTqis hie a chically quasicon ex in Xp
Tq.
P oo . 1. The i s asse ion o his i em ollows om he ac ha he p ojec ions o hy-
pe bolic spaces o elemen s in XpTqa e no modi ied by deco a ing he ee T. Fu -
he mo e, by he cons uc ion o Sec ion 3.1.1, he e exis s a cons an Cą0 such ha
C
T Vs‹“NC`π
T Vs‹pXpTqq˘, and he e o e π
T Vs‹pXpTqq is quasi-isome ic o C
T Vs‹.
2. As he a o i e ep esen a i e o he equi alence class Vs‹is he same as o he class Vs,
i ollows ha π Vs‹pXpTqq is isome ic o π VspXpTqq. The second asse ion o his i em
ollows om he equali y Xp
Tq “ NC`XpTq˘.
3. By wha was jus p o ed in he p e ious poin s, πUpXpTqq is kp0q-quasicon ex in πUpXp
Tqq,
o all UPS, o some ixed numbe kp0q.
Mo eo e , le ~
bbe a κ-consis en uple such ha bXPπXpXpTqq o e e y XPSand le
xPXp
Tqbe a ealiza ion poin o ~
b. Since Xp
Tq “ NCpXpTqq he e exis s x1PXpTqsuch
ha dXp
Tqpx, x1q ď C, and he e o e he p oo is comple e.
As al eady men ioned in Sec ion 3.1.1, o cons uc he hie a chically hype bolic s uc u e o
he g aph o hie a chically hype bolic g oups Go Co olla y 3.3.1, we do no conside di ec ly a
80 CHAPTER 3. A COMBINATION THEOREM
deco a ed ee, because he e migh no be a non- i ial ac ion o he undamen al g oup o Gon
ha hie a chically hype bolic space. Ins ead, we p oceed as ollows. Le
(3.25) T“´T, HwuwPV, H u PE, φ ˘u¯
be he ee o hie a chically hype bolic g oups associa ed o G, as desc ibed in [14, Sec ion 8.2]. In
pa icula , T“ pV, Eqis he Bass-Se e ee associa ed o he ini e g aph Γ, each Hwis conjuga ed
in he o al g oup G o G , whe e wmaps o ia he quo ien map TÑΓ, analogously H is
conjuga ed o Ge, and he edge maps φ ˘ag ee wi h hese conjuga ions o edge and e ex g oups
o gi e he embeddings in he ee o hie a chically hype bolic g oups. Le XpTqbe he associa ed
me ic space, and le Sdeno e he index se associa ed o XpTq, as desc ibed in Sec ion 3.2.
Associa ed o his, we conside he deco a ed ee
To hie a chically hype bolic g oups, as de-
sc ibed in Sec ion 3.1.1. By Theo em 3.0.1, he me ic space Xp
Tqadmi s a hie a chically hy-
pe bolic space s uc u e, ha we deno e by
S. By Lemma 2.1.15, he me ic space XpTqis
hie a chically quasicon ex in Xp
Tq, and he e o e `XpTq,
S˘is a hie a chically hype bolic space
by [14, P oposi ion 5.5], whe e he hype bolic spaces associa ed o an elemen UP
Sis de ined
as πU`XpTq˘ĎCU. F om Rema k 1.6.4, we a e assuming ha e e y πUis uni o mly coa sely
su jec i e, so in ac he e is no ha m in conside ing CUins ead o πU`XpTq˘. As
Sand Scoincide
as se s o indices (wha changes a e he hype bolic spaces associa ed o each index, as de ailed
in Sec ion 3.1.1), he abo e subs i u ion is equi alen o equipping he me ic space XpTqwi h
he hie a chically hype bolic s uc u e gi en by S. Tha is o say, `XpTq,S˘is a hie a chically
hype bolic space.
We now se o p o e Co olla y 3.3.1. Be o e showing he ull p oo we discuss how he index
se cons uc ed in Sec ion 3.2 on a ee o hie a chically hype bolic spaces can be applied o he
hie a chical hype bolic g oup s uc u e o a g aph o g oups Gon he ee o spaces ob ained by
conside ing i s Bass-Se e ee.
We i s desc ibe he hie a chical hype bolic space s uc u es in ol ed in each e ex space associ-
a ed o he ee o spaces desc ibed in Equa ion (3.25).
Rema k 3.3.3. Recall ha each e ex in he Bass-Se e ee Tco esponds o a cose gG , whe e
G is a e ex g oup co esponding o he g aph o g oups G. We endow he me ic space gG wi h
a copy o he index se S deno ed by gS such ha he e is a hie omo phism φg:pG ,S q Ñ
pgG , gS qequi a ian wi h espec o he conjuga ion isomo phism G ÑGg
. I UPS we
deno e by φpUq
g he isome y a hype bolic space le el making he ollowing diag am commu e:
G
φg//
πV

gG
πgV

CV φpV q
g
//CgV
We ecall he e he no ion o T-cohe en bijec ions, whe e Tis he ee o hie a chically hype bolic
3.3. APPLICATIONS 81
spaces. A bijec ion o he index se Sgi en in Equa ion (3.4) is said o be T-cohe en i :
•i induces bijec ions on he se s S1and S2;
•i p ese es he ela ion „on S1;
•i induces a bijec ion bo he unde lying ee T ha commu es wi h :Ů PVS ÑT, whe e
sends each VPS o he e ex . Tha is, b “b .
No ice ha he composi ion o T-cohe en bijec ions is T-cohe en . The e o e, le PTďAu pSq
be he g oup o T-cohe en bijec ions.
To p oduce he index se Sin a PT-equi a ian manne , we p oceed as ollows. No ice ha Gac s
on Ů PVS , so ha o any V PS we ha e ha g.V PSg. . This ex ends o an ac ion o S1
de ining g. Vs “ g.V s. Fo any Ws P S1, choose a le ans e sal S Wso he subg oup
S abGp Wsq “ gPG|g Ws“ Ws(,
and impose ha eGPS Ws. Fo each PT-o bi in S1choose a ep esen a i e Vso he o bi , a
a o i e e ex o Vs, and a a o i e ep esen a i e V PS o Vs. Fo any elemen gPG,
he e is a unique elemen lPS Vssuch ha gPl¨S abGp Vsq. We decla e l o be he a o i e
e ex o g Vs, and gV PSl. o be he a o i e ep esen a i e o he equi alence class g. Vs.
This de ini ion is consis en , ha is ha i g, ˜gPG, hen he a o i e e ex o pg˜gq. Vscoincides
wi h he a o i e e ex o g.`˜g. Vs˘. Indeed, suppose ha ˜gP˜
l¨S abGp Vsq, ha g˜gPp¨
S abGp Vsq, and ha gPl‹¨S abGp˜g Vsq, o unique elemen s ˜
l, p PS Vsand l‹PS˜g Vs. Thus,
he a o i e e ex o g˜g Vsis p. , and i s ep esen a i e is Vp. PSp. . On he o he hand, he
a o i e e ex o ˜g Vsis ˜
l. , wi h a o i e ep esen a i e ˜
l Vs, and consequen ly he a o i e e ex
o g`˜g Vs˘is pl‹˜
lq. , wi h a o i e ep esen a i e Vpl‹˜
lq. . As gPl‹¨S abGp˜g Vsq and S abGp˜g Vsq “
˜gS abGp Vsq˜g´1, we ha e ha g˜gP pl‹˜gq¨S abGp Vsq “ pl‹˜
lq¨S abGp Vsq. The e o e, as g˜gbelongs
o a unique cose o S abGp Vsq, we ha e ha p¨S abGp Vsq “ pl‹˜
lq¨S abGp Vsq, which implies
ha l‹˜
l Vs “ pp´1l‹˜
l Vs “ p Vs. As a consequence, he a ou i e e ices and ep esen a i es o
g g Vsand gp g Vsq a e equal.
F om he de ini ion o he ac ion o PTon S2, i ollows ha Cg.T Us“CTg. Us.
Lemma 3.3.4. Le G“`Γ, G u PV, GeuePE, φe˘:GeÑGe˘uePE˘be a ini e g aph o hie a -
chically hype bolic g oups sa is ying he hypo heses o Co olla y 3.3.1. Fu he , le Tbe he ee
o hie a chically hype bolic spaces associa ed o Gas in Equa ion (3.25). I gPG“π1pGqsuch
ha g Vs“ Ws hen o e e y PT Vsand ep esen a i e V PS o Vs he e exis an isome y
gV :CV ÑCWg. making he ollowing diag am uni o mly coa sely commu e
G
g//
πV

Gg
πWg.

CV gV
//CWg.
88 CHAPTER 3. A COMBINATION THEOREM
Fo each Vs P S1, he p ojec ion π Vs, as de ined in Equa ion (3.7) and Equa ion (3.8), is
π Vspxq “ $
’
&
’
%
cw˝πVwpxq,@xPX , PT Vs;
ce`˝πVe`pφe`pXeqq,@xPX , RT Vs,
whe e e“ep qis he las edge in he geodesic connec ing o T Vssuch ha e`PT Vs, and he
maps cwand ce`deno e he app op ia e compa ison maps o he a o i e ep esen a i e o Vs.
Le xPX ĎXand le T VsPS2. Then, πT Vspxqis de ined as he composi ion o he closes
poin p ojec ion o o T Vsin he Bass-Se e ee T, wi h he inclusion o T Vsin o he coned-
o CT Vs“p
T Vs.
To p o e ha GΓz uis hie a chically quasicon ex in GΓ, we need o check he wo condi ions o
De ini ion 1.7.4. Fo each elemen T VsPS2we ha e ha πT VspGΓz uqis a poin in CT Vs“p
T Vs
and, he e o e, i is quasicon ex in CT Vs.
Suppose ha Vs P S1, and assume ha Vshas a ep esen a i e in g.S , whe e S is he index
se associa ed o he e ex g oup G . In pa icula Vs “ Vu, and π VspGΓz uq Ď πVpg.Glinkp qq.
Since VRg.Slinkp q, he se πVpg.Glinkp qqis uni o mly bounded, and he e o e π VspGΓz uqis
quasicon ex in C Vs.
On he o he hand, assume ha he g oup o bi G. Vsin e sec s SΓz u. Wi hou loss o gene ali y,
as he g oup ac s isome ically on he hype bolic spaces, we can assume ha Vshas a ep esen-
a i e ˜
VPSΓz u. By de ini ion π VspGΓz uq “ c˝π˜
VpGΓz uq, whe e cis he compa ison map
om ˜
V o he a ou i e ep esen a i e o Vs. By Axiom (1) o De ini ion 1.6.1, he se π˜
VpGΓz uq
is quasicon ex in C˜
V, and he e o e π VspGΓz uqis quasicon ex in C Vs, being can isome y. I
ollows ha o e e y elemen Vs P S1, he se π VspGΓz uqis quasicon ex in C Vs.
To conclude he p oo o hie a chical quasicon exi y, conside a consis en uple ~
bin pG, Sqsuch
ha b VsPπ VspGΓz uqand bT VsPπT VspGΓz uq o e e y Vs P S1. The se s πT VspGΓz uqa e
uni o mly bounded, being poin s, o all T VsPS2. Mo eo e , π VspGΓz uqa e uni o mly bounded
o e e y equi alence class Vs P S1which has a ep esen a i e in g.S .
Le αdeno e he e ex o he Bass-Se e ee in which he subg oup GΓz uis suppo ed. Le
i:GΓz uÑGΓbe he hie omo phism de ined as ollows. A he me ic-space le el de ine i o be
he na u al inclusion. A he le el o index se s i♦pUq“ Usand, a he le el o hype bolic spaces,
i˚
U:CUÑC Usis he compa ison map c:CUαÑC Us, which is an isome y.
Fo each Vs P S1, we ha e ha
π VspGΓz uq “ $
’
&
’
%
cα˝πVαpGΓz uq,i αPT Vs;
ce`˝πVe`pφe`pXeqq,i αRT Vs.

3.3. APPLICATIONS 89
By Theo em 2.2.1 he se πVe`pφe`pXeqq is uni o mly bounded, and hus ce`˝πVe`pφe`pXeqq is
uni o mly bounded. Fo each Vs P S1such ha αPT Vs, le c Vsdeno e cpb Vsq, whe e he maps
cdeno e he compa ison maps (which a e isome ies) om he a ou i e ep esen a i e o Vs o
he ep esen a i e Vα( he e o e, he maps cchange wi h espec o di e en equi alence classes).
Conside he consis en uple
~c “ź
VsPS1,
αPT Vs
c Vs
By induc ion hypo hesis, GΓz uis a hie a chically hype bolic g oup. The e o e, he consis en uple
~c admi s a ealiza ion poin zPGΓz u, and hus we ob ain ha π Vspzq — b Vs o e e y Vs P S1.
Fu he mo e, since πT VspGΓz uqis a poin , we also ha e ha πT Vspzq “ bT Vs“πT VspGΓz uq o
e e y T VsPS2. Tha is, he second condi ion o hie a chical quasicon exi y is p o ed, and he
inclusion GΓz uãÑGΓis a hie a chically quasicon ex hie omo phism.
Mo eo e , o each VPSΓz u he map CVÑC Vsis an isome y. No e ha , i an elemen
VsĎi♦pUq“ Us, whe e UPSΓz u, hen T UsĎT Vs. By assump ion αPT Us, and he e o e
αPT Vsand he e exis s VPSΓz usuch ha i♦pVq “ Vs.
Thus, we p o ed ha all induc ion hypo heses a e sa is ied by he inclusion GΓz uãÑG, ha is
ha he embedding is a ull, hie a chically quasicon ex hie omo phism, which induces isome ies
a he le el o hype bolic spaces.
To deduce he same o an a bi a y G∆, we p oceed as ollows. I ∆ “Γz uu o some (o he )
e ex uPV, hen he abo e a gumen , whe e in Equa ion (3.29) we conside he spli ing o e
he subg oup Glinkpuq, p o es ha he inclusion G∆ãÑGsa is ies he desi ed p ope ies. I no ,
hen ∆ is a p ope subg aph o Γz uu, o some uPV. Induc ion p o es ha he embedding
G∆ãÑGΓz uusa is ies said p ope ies, and again he abo e a gumen p o es he claim o he
inclusion GΓz uuãÑG. As ullness, hie a chical quasicon exi y, and inducing isome ies a he le el
o hype bolic spaces, a e all p ope ies p ese ed by composi ion o hie omo phisms, we conclude
ha he inclusion G∆ãÑGsa is ies he induc i e s a emen , and he p oo is hus comple e.
We end he chap e wi h a ema k ha an icipa es wha he ollowing chap e is abou . In sho ,
i shows he limi s o applica ion o Theo em 3.0.1 o gene al g aphs o g oups.
Example 3.3.8. [Baumslag–Soli a g oups] Le us conside mo e in de ail non-euclidean Baumslag–
Soli a g oups BSp1, kq “ xa, | a ´1“aky, whe e k‰ ˘1. Le T“ pV, Eqbe he Bass–Se e
ee associa ed o he HNN ex ension BSp1, kq, so ha V“ gxay | gPBSp1, kqu. Two dis inc
e ices gxayand hxaya e joined by an edge ePEi and only i he e exis s bP xaysuch ha ei he
hxay “ gb ˘1xay, o hxay “ gb ´1xay. Fo a e ex gxay “ PVle `X ,S ˘:“`gxay, xayu˘
be he hie a chically hype bolic space associa ed o he e ex, and o any edge ePEle
`Xe,Se˘:“`xay, xayu˘be he hie a chically hype bolic space associa ed o he edge. Gi en
90 CHAPTER 3. A COMBINATION THEOREM
gxay, hxayu “ ePE, conside he hie omo phisms φe`:`xay, xayu˘Ñ`gxay, xayu˘be de ined as
φe`paq “ ga, and φe´:`xay, xayu˘Ñ`hxay, xayu˘be de ined as φe´paq “ hak.
We ha e ha
T“´T, ``Xgxay,Sgxay˘˘(gPG, `Xe,Se˘u˘(ePE, φe˘(ePE¯
is a ee o hie a chically hype bolic spaces. The e ex-spaces and edge-spaces all ha e he in-
e sec ion p ope y and clean con aine s, because hei index se consis s o only one elemen .
Mo eo e , hie omo phisms a e hie a chically quasicon ex, uni o mly coa sely lipschi z, and ull.
Le us p o e ha compa ison maps a e no uni o m quasi isome ies. Fi s no ice ha , as each
hie a chically hype bolic space has an index se o ca dinali y one, he e is only equi alence class
ha spans he whole ee T. Le and ube wo e ices in T, a dis ance d. Then, he compa ison
map c Ñu:xay Ñ xayis a p|k|d,0q-lipschi z map. The e o e, as |k|ą1 and we canno bound he
dis ance dbe ween wo e ices in he unbounded ee T, compa ison maps canno be uni o m
quasi isome ies, as claimed.
The abo e ema k shows ha Theo em 3.0.1 canno be applied o show ha non-euclidean Baum-
slag Soli a g oups a e hie a chically hype bolic. The ollowing esul is an analog o Lemma 1.2.6,
i shows ha hie a chically hype bolic g oups canno con ain in ini e dis o ed cyclic subg oups.
Rema k 3.3.9. I Gis a hie a chically hype bolic g oup, hen Gcanno ha e a subg oup isomo -
phic o BSpn, mq “ xa, | an ´1“amy, wi h |n| ‰ |m|. Indeed, suppose he e is an embedding
ι:BSpn, mqãÑG. We ha e ha ιpaqis an in ini e o de elemen o G. By [35, Theo em 7.1] and
[36, Theo em 3.1], ιpaqis undis o ed, which is a con adic ion.
Mo e gene ally, i a g oup Ghas a hie a chical hype bolic s uc u e, hen i canno be unbalanced,
as i canno con ain in ini e dis o ed cyclic subg oups.
A e examining he abo e ema k, one would be emp ed o hink ha he only way ha a
Baumslag Soli a g oup BSpm, nqhas a hie a chically hype bolic s uc u e p ecisely when |m| “
|n|. This is indeed, he case, and we de o e he las chap e o his hesis o s udy hie a chical
hype bolici y o a much b oade class o g oups ha we choose o call hype bolic-2-decomposable
g oups.
Chap e 4
Hie a chical hype bolici y o
hype bolic-2-decomposable g oups
In his chap e we will conside g oups ha spli as g aphs o g oups wi h 2-ended edge g oups.
Recall ha , i Pis a p ope y o a g oup, we say ha a g oup is P-2-decomposable i i spli s as a
g aph o g oups wi h 2-ended edge g oups and e ex g oups sa is ying p ope y P.
We now s a e he main esul o he chap e .
Theo em 4.0.1. Le Gbe a hype bolic-2-decomposable g oup. The ollowing a e equi alen .
1. Gadmi s a hie a chically hype bolic g oup s uc u e.
2. Gdoes no con ain a dis o ed in ini e cyclic subg oup.
3. Gdoes no con ain a non-Euclidean almos Baumslag–Soli a g oup.
Mo eo e , i Gis i ually o sion- ee, condi ion (3) can be eplaced by
3’. Gdoes no con ain a non-Euclidean Baumslag–Soli a g oup.
Be o e we begin wi h he chap e , we s a e a ew ques ions and possible u u e di ec ions.
4.0.1 Ques ions
The non i ually o sion- ee case: ou esul s a e s a ed di e en ly o he case o i -
ually o sion- ee g oups. The main p oblem being ha we could no de e mine in he class
o hype bolic-2-decomposable g oups whe he all non-Euclidean almos Baumslag–Soli a g oups
con ain a Baumslag–Soli a subg oup.
Ques ion 4.0.2. Does e e y non-Euclidean almos Baumslag–Soli a subg oup o a hype bolic-2-
decomposable g oup con ain a non-Euclidean Baumslag–Soli a subg oup?
91
92 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
We s ess ha his ques ion has a posi i e answe o ce ain o sion- ee g oups. In [59, P oposi-
ion 7.5] he au ho shows ha he ques ion has a posi i e answe o GBS g oups. In [28, P opo-
si ion 9.6] he au ho ex ends he esul o ( o sion- ee hype bolic)-2-decomposable g oups. How-
e e , he esul s appea ing in hose pape s ely hea ily on he absence o o sion. As we will see in
Sec ion 4.1, i is enough o assume ha Gis i ually o sion- ee. Mo eo e , ecall ha a g aph
o i ually o sion- ee g oups may no ha e a i ually o sion- ee undamen al g oup (Example
1.9.16).
Gene aliza ion o HHG-2-decomposable In ou p oo s, hype bolici y o he edge g oups is
used only in Theo em 4.2.2 and Lemma 4.2.6. Thus we expec ha inding app op ia e eplace-
men s o he wo esul s abo e will yield a su icien condi ion o a (hie a chically hype bolic)-2-
decomposable g oup o be hie a chically hype bolic. Howe e , he ques ion becomes ha de when
asking o a ull cha ac e iza ion. As ema ked be o e, all hie a chically hype bolic g oups a e
balanced, hence balancedness is su ely a necessa y condi ion in Ques ion 2.
Ques ion 4.0.3. Unde which condi ions a (hie a chically hype bolic)-2-decomposable g oup is
hie a chically hype bolic?
A possible s a egy o answe his ques ion would be o ex end he ools de eloped in Sec ion 4.2
o he class o hie a chically hype bolic g oups. Tha is o say, p o ide condi ions gua an eeing
ha he hie a chically hype bolic s uc u e o edge g oups can be included in he one o he e ex
g oup.
Howe e , we don’ hink his s a egy would wo k in he gene al case. Fo ins ance, conside Z2-
2-decomposable g oups (also known as ubula g oups). I one e ex has h ee incoming edges,
de ining pai wise linea ly independen lines, he e is no s aigh o wa d way o de ining a hie a -
chically hype bolic g oup s uc u e on Z2 ha con ains each edge g oup.
4.0.2 Balanced g oups
A undamen al no ion h oughou he chap e is he no ion o balanced g oup.
De ini ion 4.0.4. Le Gbe a g oup and gPG. We say ha gis balanced ei he i ghas ini e
o de , o i whene e gnis conjuga e o gm, i mus ollow |n|“|m|. We say ha a g oup Gis
balanced i e e y elemen is balanced.
Lemma 4.0.5 ([90, Lemma 4.14]).Le Gbe a g oup and assume ha he e exis s a balanced
subg oup Ho Go ini e index. Then, Gis balanced.
We a e now going o s udy how balanced g oups beha e unde amalgama ed p oduc s and HNN
ex ension o e i ually cyclic g oups. A key p ope y o i ually cyclic g oups ha will be used
h oughou he chap e is ha i a, b a e in ini e o de elemen s o a i ually cyclic g oup, hen
he e a e N, M such ha aN“bM.
Lemma 4.0.6. Le Cbe a i ually cyclic g oup and G“A˚CB. Then Gis balanced i and only
i A, B a e.
93
P oo . One implica ion is clea . To show he con e se, le gPGbe an in ini e o de elemen and
le hPGbe such ha hgnh´1“gm o |n|‰|m|. I gis ac s hype bolically on he Bass-Se e ee
Tco esponding o G, hen he ansla ion leng h `Gpgqis posi i e. Mo eo e , `Gpgnq“|n|`Gpgq
and `Gphgh´1q “ `Gpgq. Thus, i hgnh´1“gm hen |n|“|m|, which is a con adic ion. Thus, we
can assume ha gac s ellip ically on T.
The e o e, he e exis s xsuch ha xgx´1belongs in Ao B. Assume wi hou loss o gene ali y
ha xgx´1PA. We ha e
(4.1) pxhx´1qpxgx´1qnpxhx´1q´1“ pxgx´1qm.
I we w i e a“ pxgx´1q P Aand k“xhx´1, Equa ion (4.1) becomes kank´1“am. W i e kin
no mal o m k0¨¨¨ks, whe e kiPA´1 o B´1. We ha e
pk0¨¨¨ksqaT npk0¨¨¨ksq´1a´T m “1.
The e a e now wo cases. Fi s , assume ha no powe s o acan be conjuga ed in o C, o ins ance,
his happens whene e |C|ď8. Then by he no mal o m heo em, s“0k0PAand hence A
was no balanced.
So suppose ha he e is some powe ao a ha can be conjuga ed in o C. Up o conjuga ing a
and kand aking powe s o a, we can assume ha aPCand kank´1“amholds. Again, conside
he no mal o m k“k0. . . ks. We will p oceed by induc ion on s.
Case s“0. In his case we ha e k0ank´1
0“am. Since aPC, i k0PA( esp. B), we ha e ha A
( esp. B) is unbalanced.
Induc ion s ep. Suppose ha he claim holds o kwi h no mal- o m leng h s´1. We will show
ha i holds o leng h s. Conside he equa ion kank´1“amand assume ha khas no mal- o m
leng h s. Obse e ha o each T he equa ion kaT nk´1“aT m s ill holds. We will show ha , o
Tla ge enough, we can w i e kaT nk´1“aT m as k1cn1pk1q´1“cm1wi h cPC,|n1| ‰ |m1|and k1
wi h no mal- o m leng h a mos s´1. Then we a e done by induc ion hypo hesis.
We ha e
pk0¨¨¨ksqanpk0¨¨¨ksq´1“am.
By he no mal o m heo em, b“ksank´1
sPC. Since Cis 2-ended, he e is cPCand P1, P2, P3, P4
such ha aP1“cP2and bP3“cP4. Le K“k0¨¨¨ks´1. Then we ha e
(4.2) KksaP1P3nk´1
sK´1“aP1P3m

94 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
Le ’s ocus on he le -hand side only, conjuga ing i by K. We ha e
kscP2P3nk´1
s“ksaP1P3nk´1
s“bP1P3“cP1P4.
Since ksbelongs o ei he Ao B, all he elemen s o he abo e se ies o equa ions a e in one
be ween A, B, say A. Since Ais balanced, we need o ha e |P2P3n|“|P1P4|. Thus, up o possibly
subs i u ing nwi h ´n, we can w i e he le -hand-side o Equa ion (4.2) as KcP2P3nK´1. Now,
applying he equali y aP1“cP2 o he igh -hand-side o Equa ion (4.2), we ha e
KcP1P4K´1“KcP2P3nK´1“cP2P3m.
We a e now done by induc ion hypo hesis.
By applying epea edly he p e ious lemma, we ob ain he ollowing co olla y.
Co olla y 4.0.7. I Gis a balanced-2-decomposable g oup such ha he unde lying g aph is a ee,
hen Gis balanced.
I is s aigh o wa d o check ha HNN ex ensions o balanced g oups a e no balanced in gene al:
Simply conside BSp2,3qas he HNN ex ension xa, | a2 ´1“a3y – xay˚ a2 ´1“a3.
To inish his subsec ion we include esul s ha gi e su icien condi ions o an HNN ex ension
o e a balanced g oup o be balanced. We s ess ha hese esul s a e modi ied e sions o
[28, P oposi ion 6.3] and [28, Theo em 6.4]. They ha e been modi ied as o allow o sion.
P oposi ion 4.0.8. Le Hbe a balanced g oup, A, B ďHbe i ually cyclic subg oups and
φ:AÑBbe a isomo phism. Le G“H˚φ. Then.
1. I gPHbu no powe o gis conjuga e in Hin o AYB hen gis s ill balanced in G.
2. I Aand Ba e non-commensu able in H, hen Gis also a balanced g oup.
P oo . Suppose gwas no balanced in G. Hence he e is hPG´Hsuch ha hgph´1“gq o some
|p| ‰ |q|. Since hPG´H, we can w i e h“h1 ε1. . . h ´1 ε h in educed o m. By assump ion
h gh´1
does no belong o Ano B, and hence hgqh´1canno ep esen an elemen o H. Thus,
hPHand since His balanced |q|“|p|.
Fo he second i em, we only need o check he balancedeness o ellip ic elemen s in G, since a
ansla ion leng h a gumen simila o ha o Lemma 4.0.6 ules ou unbalancedeness o hype bolic
elemen s. Thus, i Gis unbalanced, by he i s i em he e mus exis an unbalanced in ini e o de
elemen hPHsuch ha some powe o hcan be conjuga ed in o AYB. The e o e, we can assume
wi hou loss o gene ali y ha hPAYB. Assume ha h“aPA. Since ais unbalanced, he e
is some gPGsuch ha gaig´1“ajwi h |i|‰|j|. Le g“h1 ε1. . . h ε be he educed o m
95
exp ession in G. Since ghig´1“hjhas no mal o m leng h 1, he e mus exis some possible
educ ion in ph1 ε1. . . h ε qhiph1 ε1. . . h ε q´1. The e a e wo possible ways ha his could
happen: Ei he ε “1 and h hih´1
PAo ε “ ´1 and h hih´1
PB. I he la e occu s,
hen he p oo is comple e, as h hih´1
is an in ini e o de elemen in Ah XB. Assume now
ha he o me case occu s. Since Ais a 2-ended balanced g oup, he e mus exis ksuch ha
h aikh´1
“a˘ik. The e o e, ε h aikh´1
´ε “ a˘ik ´1“b˘ik. Again, as be o e, we ha e wo
possibili ies: Ei he h ´1b˘ikh´1
´1belongs in Band ε ´1“ ´1 o h ´1b˘ikh´1
´1belongs in Aand
ε ´1“1. I he la e occu s, he p oo is comple e. I he o me occu s, since Bis a 2-ended
balanced g oup, hen h ´1b˘ikk1h´1
´1“b˘ikk1 o some k1. We can con inue pe o ming educ ions
in he exp ession o gaig´1and a each s ep we ha e he same dicho omy whe e ei he he p oo is
comple e o we can con inue educing. No e ha a some poin o he educ ion we ob ain hisuch
ha AhiXBo AXBhiis in ini e. Indeed, o he wise o some K‰0 he equali y gaKig´1“aKj
would hold o |Ki|“|Kj|, con adic ing he assump ion.
Co olla y 4.0.9. Le Gbe an HNN ex ension o he balanced g oup Hwi h s able le e and
2-ended associa ed subg oups Aand Bo H. Le aPA, b PBbe in ini e o de elemen s such ha
a ´1“b. Mo eo e , suppose ha he e is hPHconjuga ing a powe o a o a powe o b, so
ha haih´1“bj. Then Gis balanced i and only o e e y pai o elemen s a, b as abo e we ha e
|i|“|j|.
P oo . One implica ion is clea , we now show ha Gis balanced p o ided ha o e e y hPH
such ha haih´1“bj o some i, j i ollows ha |i|“|j|.
Assume ha Gis an unbalanced g oup. The e o e, by he second asse ion in he p e ious p opo-
si ion, he e mus exis some h1PHsuch ha AXh1Bh1´1is in ini e. Since HNN ex ensions
a e de ined up o conjuga ion o he co esponding embedding maps, by conjuga ing by h1we can
assume ha AXBis in ini e in H. By he i s asse ion in he p e ious p oposi ion, he only
elemen s ha can be unbalanced a e hose hPH ha can be conjuga e in Hin o AYB. Thus,
we can assume wi hou loss o gene ali y ha he unbalanced elemen s in Gbelong in AYB.
The e o e, i Gis unbalanced, we can assume ha o some aPA he e is some gPGsuch ha
gang´1“am o some |n| ‰ |m|. We will induc on he leng h o he educed o m o g o show
ha gang´1“amimplies |n|“|m|, ob aining a con adic ion.
Le g“h0 ε1h1. . . ε h be he educed exp ession o g. Le us say ha deno es he educed o m
leng h o g. Assume ha “0. Tha is o say, gPH. Since His balanced, we ha e |n|“|m|.
Assume now ha he claim holds o elemen s o educed o m leng h ´1, and le go educed
o m leng h be such ha gang´1“am. We deno e by bPB he elemen such ha a ´1“b.
No e ha i he equa ion gang´1“amholds in G, hen o e e y Twe ha e ha gaT ng´1“aT m
o e e y Tą0. Since he elemen gang´1“ambelongs in H, by he no mal o m heo em,
96 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
gang´1mus admi some educ ion in i s educed o m. The e a e wo ways ha his educ ion
can occu : Ei he ε “1 and h anh´1
belongs in Ao ε1“ ´1 and h anh´1
belongs in B.
No e ha in he o me case, since Ais 2-ended and balanced, he e mus exis some ksuch
ha h aknh´1
“a˘kn. The e o e, ε
h aknh´1
´ε
“b˘kn. In he la e case we ha e ha
h anh´1
“b1PB. Since Bis a 2-ended g oup, he e mus exis l1, l2such ha pb1ql1“bl2. Thus,
h anl1h´1
“ pb1ql1“bl2. By assump ion, we mus ha e ha |nl1| “ |l2|. The e o e, in he la e
case we ha e ha ´1h anl1h´1
“ ´1b˘l2 “a˘l2“a˘nl1. In bo h cases, we use he induc ion
s ep o conclude |kn| “ |km|o |l1n| “ |l1m| espec i ely. In pa icula , since k‰0‰l1, we
conclude |n|“|m|.
4.0.3 Con exi y
In his chap e , we will make use o wo no ions o con exi y. The i s one, called hie a chical
quasicon exi y, hea ily elies on he hie a chical s uc u e. Fo ins ance, i is no quasi-isome ic
in a ian . Fo a mo e p ecise accoun , we e e o [73].
To de ec hie a chical quasicon exi y some imes i is con enien o check a s onge p ope y.
De ini ion 4.0.10 (S ong quasicon exi y). A subse Yo a quasigeodesic space Xis said o
be s ongly quasicon ex i he e is a unc ion M: 1,8q Ñ Rsuch ha e e y λ–quasigeodesic in
Xwi h endpoin s in Ys ays Mpλq–close o Y.
Theo em 4.0.11 ([73, Theo em 6.3]).Le pG, Sqbe a hie a chically hype bolic g oup and YĎG
be a subse . Then i Yis s ongly quasicon ex, i is hie a chically quasicon ex, whe e he cons an s
de e mine each o he .
A special case o s ongly quasicon ex subse s is gi en by pe iphe al subg oups o ela i ely hy-
pe bolic g oups.
Lemma 4.0.12 ([34, Lemma 4.15]).Le Pbe a pe iphe al subg oup in he ela i ely hype bolic
g oup G. Then Pis s ongly quasicon ex.
In he case o hype bolic spaces, ela i e hype bolici y and s ong quasi-con exi y a e in ima ely
ela ed.
De ini ion 4.0.13. We say ha a collec ion o subg oups Hiun
i“1o Gis almos -malno mal i
HiXgHjg´1is ini e unless i“jand gPHi.
Theo em 4.0.14 ([22, Theo em 7.11]).Le Gbe a hype bolic g oup and Hiun
i“1be a ini e amily
o subg oups o G. Then Gis hype bolic ela i e o Hiui and only i Hiuis an almos -malno mal
amily o s ongly quasicon ex subg oups.
De ini ion 4.0.15 (Glueing hie omo phism). Le pH, S1qand pG, S2qbe hie a chically hy-
pe bolic g oups. A glueing hie omo phism be ween Hand Gis a g oup homomo phism φ:HÑG
4.1. HIERARCHICAL HYPERBOLICITY OF (2-ENDED)-2-DECOMPOSABLE GROUPS 97
ha can be ealized as a ull hie omo phism pφ, φ♦, φ˚
Uqsuch ha he image φpHqis hie a chically
quasi-con ex in Gand he maps φ˚
U:CUÑCφ♦Ua e isome ies o each UPS1. I he map
φ:HÑGis injec i e, we say ha he glueing hie omo phism is injec i e.
4.1 Hie a chical hype bolici y o (2-ended)-2-decomposable
g oups
In his sec ion, we ocus on (2-ended)-2-decomposable g oups. Tha is o say, g aphs o g oups
whe e e e y e ex and edge g oup is 2-ended. We begin he sec ion by ecalling some use ul esul s
on 2-ended g oups.
4.1.1 Two-ended g oups
In his subsec ion, we ecall basic esul s and ema ks on he s uc u e o wo-ended g oups. An
impo an esul o hese ype o g oups is known as he s uc u e heo em o in ini e i ually
cyclic g oups. Th oughou he chap e , we will make use o his ac on many occasions.
Lemma 4.1.1 ([89, Lemma 4.1]).I Gis an in ini e i ually cyclic g oup, hen ei he
1. Gadmi s a su jec ion wi h ini e ke nel on o he in ini e cyclic g oup Z, o
2. Gadmi s a su jec ion wi h ini e ke nel on o he in ini e dihed al g oup D8
We ecall ha he in ini e dihed al g oup is he g oup de ined by he p esen a ion D8“ x , s |
s s “ ´1, s2y. No e ha e e y elemen o D8can be w i en as s k, o P 0,1uand kPZ.
Mo eo e , e e y elemen o he o m s khas o de 2, and an elemen o he o m khas in ini e
o de p ecisely when k‰0. Using hose obse a ions, we ha e he ollowing Lemma.
Lemma 4.1.2. Le Gbe a i ually cyclic g oup. Le Φ1,Φ2:GÑD8be homomo phisms wi h
ini e ke nel and ini e index image. Then Ke pΦ1q “ Ke pΦ2q.
P oo . As be o e, D8“ xa, b |bab “a´1, b2y. Suppose ha he e is gPGsuch ha gPKe pΦ1q
and gRKe pΦ2q. Since gPKe pΦ1q, we conclude ha ghas ini e o de , o he wise |Ke pΦ1q|“ 8.
Since Φ2pGqhas ini e index in D8 he e exis s cPGsuch ha Φ2pcqhas in ini e o de . In
pa icula he e exis k1PZ, k2PZ´ 0usuch ha Φ2pgq “ bak1and Φ2pcq “ ak2, and so
Φ2pgcq “ bak1`k2. Again, gc has o ha e ini e o de o no con adic |Ke pΦ2q|ă 8 . Howe e ,
since gPKe pΦ1qwe ha e ha Φ1pgcq “ Φ1pcq, and so gc canno ha e ini e o de . F om his we
conclude Ke pΦ1q Ď Ke pΦ2q. The symme ic a gumen yields he claim.
Rema k 4.1.3. No e ha an in ini e i ually cyclic g oup Gcanno su jec on o bo h Zand D8
wi h ini e ke nel. Indeed, assume ha wo su jec i e homomo phisms Φ : GÑZand Φ1:GÑD8
104 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
2. i π1pGqis i ually o sion- ee hen π1pGqmus con ain a non-Euclidean Baumslag-Soli a
subg oup.
P oo . By de ini ion o balanced edges (De ini ion 4.1.15), i eis unbalanced and φ˘a e he
monomo phisms associa ed o he edge e, hen he e exis s an in ini e o de elemen a1PGe
and hPπ1pG´eqsuch ha hφ`pa1qih´1“φ´pa1qj o some |i|‰|j|. Le adeno e φ`pa1q
and sdeno e eh o sho . By assump ion, ahas in ini e o de , and so s‰1. Then xa, syis a
non-Euclidean almos Baumslag-Soli a g oup.
I , in addi ion, π1pGqis i ually o sion- ee hen he e exis s Ną1 such ha aNand sNbelongs
in a o sion- ee subg oup o π1pGq. No e ha
sNaN¨iNs´N“sN´1pspaiqN¨iN´1s´1qs´pN´1q“
“sN´1ppajqN¨iN´1qs´pN´1q“
“sN´2pspaiqJN¨iN´2s´1qs´pN´2q“
“ ¨¨¨ “ aN¨jN
The e o e, he ela ion sNpaNiNqs´N“aNjNis sa is ied in a o sion- ee subg oup Qo π1pGq.
By Lemma 4.1.5, Qis a gene alized Baumslag-Soli a g oup. Since NiN{NjN“ pi{jqN‰ ˘1,
by [59, P oposi ion 7.5] he subg oup xaN, sNycon ains some non-Euclidean Baumslag-Soli a
g oup.
Combining Lemma 4.1.19 wi h Co olla y 4.1.22 we ob ain Theo em 1.9.15 om he in oduc ion:
Theo em 4.1.23. Le Gbe a g aph o g oups whe e none o he e ex g oups con ain dis o ed
cyclic subg oups. Then π1pGqcon ains a non-Euclidean almos Baumslag-Soli a subg oups i and
only i Ghas an unbalanced edge.
P oo . I G“π1pGqcon ains a non-Euclidean almos Baumslag-Soli a subg oup hen i is unbal-
anced. By Lemma 4.1.19 we ob ain ha Gmus con ain some unbalanced edge. Co olla y 4.1.22
shows he con e se.
We a e now eady o p o e he main esul o his sec ion.
Theo em 4.1.24. Le Gbe a g aph o g oups, whe e all e ex and edge g oups a e wo-ended.
Assume mo eo e ha π1pGqis i ually o sion- ee. Then he ollowing a e equi alen .
1. π1pGqadmi s a hie a chically hype bolic g oups s uc u e.
2. Gis linea ly pa ame izable.
3. π1pGqis balanced.

4.1. HIERARCHICAL HYPERBOLICITY OF (2-ENDED)-2-DECOMPOSABLE GROUPS 105
4. π1pGqdoes no con ain BSpm, nqwi h |m|‰|n|.
5. π1pGqdoes no con ain a dis o ed in ini e cyclic subg oup.
P oo .
3ô2 By Co olla y 4.1.18 we ha e ha π1pGqis linea ly pa ame izable i and only i e e y edge
ein Gis balanced. Mo eo e , by Lemma 4.1.19 we ha e ha e e y edge in Gis balanced i and
only i π1pGqis balanced.
5ñ3 Assume ha π1pGqis unbalanced. The e o e, by Lemma 4.1.19 he e is an edge e, an
in ini e o de elemen aPGeand an elemen hPπ1pG´eqsuch ha
hφ`paqih´1“φ´paqj,
wi h |i| ‰ |j|. Le x“φ`paqand y“φ´paq. Since eis unbalanced, he e is a spanning ee ha
does no con ain e. In pa icula , we can assume he e is a s able le e associa ed o he edge e
such ha y ´1“x. We claim ha xxyis dis o ed. No e ha xis o in ini e o de . To simply
no a ion, we will w i e A« Bi |A´B| ď . We ha e:
d`1, xN¨i˘«2|h|d`1, hxN¨ih´1˘“d`1, yN¨j˘«2| |d`1, xN¨j˘.
This is o say, o each Nwe ha e ˇˇd`1, xN¨i˘´d`1, xN¨j˘ˇˇď2p|h|`| |q. Since |i|‰|j|, i is now
a s anda d a gumen o show ha xxyis dis o ed. Indeed, es a ing he a gumen be o e o a
gene al exponen Mwe ha e d´xM, x |j|
|i|Mu¯ď |h|`| |`i. Assuming ha |i|ą|j|, we can i e a e
he inequali y abo e o ob ain ha dp1, XMqis compa able o log|j|
|i|pMq¨p|h|`| | ` iq. Tha is
o say, dp1, XMqg ows loga i hmically, showing ha he map nÞÑ xncanno be a quasi-isome ic
embedding.
4ñ3 Assume ha π1pGqis unbalanced. The e o e, by Lemma 4.1.19, Gmus con ain an unbal-
anced edge. The second i em o Co olla y 4.1.22 concludes he p oo .
1ñ5 Follows om [35, Theo em 7.1] and [36, Theo em 3.1].
2ñ1 Follows om Theo em 4.1.12.
5ñ4 Since non-Euclidean Baumslag-Soli a g oups con ains dis o ed cyclic subg oups i Gcon-
ains some non-Euclidean Baumslag-Soli a subg oup we ob ain he esul .
Theo em 4.1.25. Le Gbe a g aph o g oups, whe e all e ex and edge g oups a e wo-ended.
Then he ollowing a e equi alen .
1. π1pGqadmi s a hie a chically hype bolic g oups s uc u e.
106 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
2. Gis linea ly pa ame ized.
3. π1pGqis balanced.
4. π1pGqdoes no con ain a non-Euclidean almos Baumslag-Soli a subg oup.
5. π1pGqdoes no con ain a dis o ed in ini e cyclic subg oup.
P oo . Assume ha π1pGqis unbalanced. The e o e, by Lemma 4.1.19, Gmus con ain an un-
balanced edge. The i s i em o Co olla y 4.1.22 shows he implica ion 4 ñ3. The es o he
implica ions a e he same as in Theo em 4.1.24.
4.2 Hie a chical hype bolici y o hype bolic-2-decomposable
g oups
In his sec ion, we gi e a necessa y and su icien condi ion o he undamen al g oup o a g aph
o g oups wi h hype bolic e ex g oups and i ually cyclic edge g oups o be a hie a chically
hype bolic g oup. We do so by ex ending he ools in oduced in he p e ious sec ion. To ha
end, we make use o Theo em 4.2.2 o induce a hie a chically hype bolic g oup s uc u e on he
g oups G .
We begin by showing he ollowing lemma. This allows us, wi hou loss o gene ali y, o es ic
ou a en ion o g aphs o hype bolic g oups wi h in ini e i ually edge g oups.
Lemma 4.2.1 (Dealing wi h ini e e ices/edges). Le Gbe a g aph o g oups such ha
π1pGqis in ini e and Ghas hype bolic e ex g oups and i ually cyclic edge g oups. Then he e
exis s a ini e g aph o g oups G1wi h in ini e hype bolic e ex g oups and 2-ended edge g oups
such ha π1pG1q “ π1pGq.
P oo . Gi en a g aph o g oups Hle FpHqbe he se o edges wi h ini e associa ed edge g oup,
ha is ePEpHq||Ge| ď 8u. Le G0“G. We will p oduce a sequence o g aph o g oups
Gisuch ha π1pGiq – π1pGq,Gihas hype bolic e ex g oups and i ually cyclic edge g oups
and |FpGiq| ă |FpGi´1q|. Since he g aph o g oups is ini e, e en ually we will ind Gnsuch ha
FpGnq“H. In pa icula , i Gnhas a leas one edge, hen he associa ed edge g oup is in ini e.
Hence, he e ex g oups needs o be in ini e and we a e done. I he e a e no edges, hen he e is
a single e ex labelled by π1pGq, which is hype bolic by cons uc ion. Since, by assump ion π1pGq
is in ini e, we a e done.
Suppose Giis de ined. Fi s ly, suppose ha he e is ePFpGiqsuch ha he e exis s a spanning ee
Teo Gicon aining e( ecall ha π1pGqdoes no depend on he choice o spanning ee, as poin ed
ou in Rema k 1.3.4). Then he subg oup Ge`˚GeGe´is hype bolic by Theo em [17, Co olla y
Sec ion 7]. Then le Gi`1be de ined om Giby eplacing he edge eand he inciden e ices by
4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS107
a single e ex wi h associa ed g oup Ge`˚GeGe´, and lea ing he o he edge maps unchanged.
By doing his, we s ill ha e hype bolic e ex g oups and i ually cyclic edge g oups.
So, suppose ha no elemen o FpGiqcan be included in a spanning ee. This is o say ha
all elemen s o FpGiqa e loops. Le ePFpGiq, and le be he e ex inciden o i . Then by
[18, Co olla y 2.3], he HNN ex esion G ˚Geis hype bolic. Then we de ine Gi`1as he g aph o
g oups ob ained om Giby emo ing he edge eand changing he e ex g oup o o G ˚Ge.
F om now on, whene e we s a e a esul on a g aph o hype bolic g oups Gwe will always assume
ha he associa ed edge g oups Gea e i ually cyclic and in ini e. In o he wo ds, om now on
we assume ha he g oups conside ed a e hype bolic-2-decomposable.
Gi en a e ex g oup G , one o he main challenges ha we ha e o ace in his se ing is he
ac ha he incoming edge g oups do no necessa ily o m an almos -malno mal collec ion in G
(De ini ion 4.0.13). As a consequence, hese edge g oups may no be geome ically sepa a ed so as
o include hem in he hie a chical hype bolic s uc u e o G . The ollowing heo em sol es his
p oblem, and i is pi o al in he p oo o he main heo em in his sec ion. We also s ess ha i
is a consequence o [14, Theo em 9.1].
Theo em 4.2.2. Le Gbe a g oup hype bolic ela i e o a amily o hie a chically hype bolic g oups
pHi,Siqun
i“1. Suppose ha he e is a ini e amily o subg oups KαuαPΛand homomo phisms
φα:KαÑGsuch ha o each α he e exis s iand gPGsuch ha φαpKαqhas ini e index in
Hg
i. Finally, suppose ha each g oup Kαis equipped wi h a hie a chically hype bolic s uc u e Kα
such ha φg´1
α:pKα,KαqÑpHi,Siqis a glueing hie omo phism.
Then he e is a hie a chically hype bolic s uc u e pG, Sqon Gsuch ha φαis a glueing hie omo -
phism o e e y α. Mo eo e , i all pHi,Siqsa is y he in e sec ion p ope y, so does pG, Sq, and
simila ly o clean con aine s.
P oo . This heo em is an adap a ion o Theo em 4.2.2. We will ollow almos e ba im he pa
o he p oo ha desc ibes such a s uc u e on G, bu we will no e i y he axioms as i will no
add cla i y o he cu en p oo . We will conclude he p oo by showing ha he maps φαcan be
ealized as glueing hie omo phisms.
The s uc u e: Fo each i“1...,n and each le cose o Hiin G, ix a ep esen a i e gHi. Le
gSibe a copy o Siwi h i s associa ed hype bolic spaces and p ojec ions in such a way ha he e
is a hie omo phism HiÑgHiequi a ian wi h espec o he conjuga ion isomo phism HiÑHg
i.
Le p
Gbe he hype bolic space ob ained by coning-o Gwi h espec o he pe iphe als Hiu,
and le S“ p
GuYŮgPgŮiSgHi. The ela ion o nes ing, o hogonali y o ans e sali y be ween
hype bolic spaces belonging o he same copy SgHia e he same as in SHi. Fu he , i U, V belong
in wo di e en copies o di e en cose s, hen we impose ans e sali y be ween hem. Finally, o
e e y UPSgHiwe decla e ha Uis nes ed in o p
G.
The p ojec ions a e de ined as ollows: πp
G:GÑp
Gis he inclusion, which is coa sely su jec i e
108 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
and hence has quasicon ex image. Fo each UPSgHi, le ggHi:GÑgHibe he closes -poin
p ojec ion on o gHiand le πG
U“πHi
U˝ggHi, o ex end he domain o πU om gHi o G. Since
each πHi
Uwas coa sely Lipschi z on CUwi h quasicon ex image, and he closes -poin p ojec ion in
Gis uni o mly coa sely Lipschi z (Lemma 1.4.6), he p ojec ion πG
Uis uni o mly coa sely Lipschi z
and has quasicon ex image. Fo each U, V PSgHi, he a ious ρV
Uand ρU
Va e al eady de ined. I
UPSgHiand VPSg1Hj, hen ρU
V“πVpgg1HjpgHiqq. Finally, o U‰p
G, we de ine ρU
p
G o be he
cone-poin o e he unique gHiwi h UPSgHi, and ρp
G
U:p
GÑCUis de ined as ollows: o xPG,
le ρp
G
Upxq “ πG
Upxq. I xPp
Gis a cone poin o e g1Hj‰gHi, le ρp
G
Upxq “ ρSg1Hj
U, whe e Sg1Hjis
he Ď–maximal elemen o Sg1Hj. The cone-poin o e gHimay be sen anywhe e in CU.
By [14, Theo em 9.1], he cons uc ion abo e endows pG, Sqwi h a hie a chically hype bolic g oup
s uc u e.
Hie omo phisms: Fix α. By assump ion he e exis s iand gPGsuch ha φαpKαq Ď Hg
i.
Mo eo e , Φα“φg´1
α:pKα,Kαq Ñ pHi,Siqis a glueing hie omo phism. Ou goal is o show ha
φ:pKα,KαqÑpG, Sqcan be equipped wi h a glueing hie omo pism s uc u e.
To simpli y no a ion we will d op he αand isubsc ip and deno e pK, Kq“pKα,Kαq,φ“φα,
pH, SHq“pHi,Siqand so on.
Fo e e y VPK, de ine φ♦pVq “ gΦ♦pVqand φ˚
V“g˚˝Φ˚
V, whe e g˚is he isome y associa ed
o he mul iplica ion gPG. By assump ion, he maps Φ˚
V:CVÑCΦ♦Va e isome ies, and o
each UPSH, he space CHUand he space CGgU a e isome ic. Thus, he maps φ˚
Va e isome ies.
We need o show ha he ollowing wo diag ams coa sely commu e.
Kφ//
πK
V

G
πG
φ♦pVq

CVφ˚
U
//Cφ♦pVq
CVφ˚
V//
ρV
U

Cφ♦pVq
ρφ♦pVq
φ♦pUq

CUφ˚
U
//Cφ♦pUq
This is a ma e o unwinding he de ini ions. We will check he i s one, he second is analogous.
So, le xPK. Recall ha φpxq “ gΦpxqg´1PgHig´1. Then
πG
φ♦pVqpφpxqq “ g˚˝πHi
Φ♦pVq˝g´1˝ “ g˚˝πHi
Φ♦pVqpggHipΦpxqg´1qq.
(4.5)
No e ha dpΦpxqg´1, gHiq ď |g|. Since all he map a e coa sely Lipschi z, he e is a uni o m bound
be ween πHi
Φ♦pVqpggHipΦpxqg´1qq and πHi
Φ♦pVqpΦpxqq. Tha is, up o a uni o mly bounded e o , we
can w i e Equa ion 4.5 as
(4.6) πG
φ♦pVqpφpxqq “ g˚´πHi
Φ♦pVqpΦpxqq¯.
4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS109
On he o he hand, we ha e
(4.7) φ˚
V˝πK
Vpxq “ g˚`Φ˚
U˝πK
Vpxq˘.
Since g˚is an isome y, Equa ions (4.6) and (4.7) gi e he esul . No e ha he cons an o he
coa se commu a i i y depend on g. Howe e , since he e a e only ini ely many pai s pKα, Hiq,
we ob ain uni o mi y. Hence, he map φcan be equipped wi h a hie omo phism s uc u e. By
cons uc ion, he maps φ˚
Ua e isome ies, and he hie omo phism is ull. To see ha i has
hie a chically quasicon ex image, obse e ha i s image is a ini e Hausdo dis ance om a
pe iphe al subg oup, hence i is s ongly quasicon ex (Lemma 4.0.12). Then i is hie a chically
quasicon ex by Theo em 4.0.11. [73, Tho em 6.3].
In e sec ion p ope y and clean con aine s: We s a by checking clean con aine s, ha is
o check ha o each UĎTPSwe ha e UKcon T
KU. I U“p
G he e is no hing o check. Hence,
assume UPgSiand le gSibe he Ď–maximal elemen o gSi. Recall ha he ela ions on Sa e
de ined such ha i U, V PS´ p
Gua e no ans e se, hen he e is iP 1, . . . , nuand gPGsuch
ha U, V PgSi. In pa icula , UKVimplies U, V PgSi. Hence, con p
G
KU“con gSi
KU. Mo eo e ,
i UĎTand T‰p
G, i ollows TPgSi. Since we assumed ha pHi,Siqhas clean con aine s, we
ha e UKcon T
KU o all TPgSi, comple ing he p oo .
Conside now he in e sec ion p ope y. By hypo hesis, o each gSi he map ^gHiis de ined.
Then de ine ^:pSY HuqˆpSY Huq Ñ pSY Huq by conside ing he symme ic closu e o he
ollowing:
U^V“$
’
’
’
’
&
’
’
’
’
%
Ui V“p
G
U^gHiVi U, V PgSi o some i, g
Ho he wise.
The only p ope y o e i y ha does no ollow di ec ly is o check ha i UPgSiand VPg1Sj
wi h gSi‰g1Sj, hen he e is no Wnes ed in bo h U, V . Bu i such a Wexis ed, hen i needs
o belong o bo h gSiand g1Sj, a con adic ion.
4.2.1 Commensu abili y and conjugacy g aph
In his subsec ion we ex end he esul s ob ained in Sec ion 4.1 o he gene al se ing. The key
objec ha will allow us o do his is he conjugacy g aph (De ini ion 4.2.10). This is a g aph o
g oups ha , combined wi h Theo em 4.2.2, p o ides e ex g oups wi h a hie a chical hype bolic
s uc u e ealizing edge maps as glueing hie omo phisms.
As he e ex g oups in he g aphs o g oups conside ed a e no 2-ended, he whole g aph o g oups
canno be linea ly pa ame ized. Mo eo e , he edge g oups do no necessa ily embed in o e ex
g oups in an almos malno mal way. To o e come hose p oblems, we will conside he elemen a y

110 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
closu e o subg oups. A sys ema ic s udy o elemen a y closu es o WPD subg oups (which include
cyclic subg oups o hype bolic g oups as a special case) is ca ied on in [30], whe e he au ho s
show such subg oups needs o be hype bolically embedded in he ambien g oup. Fo he sake o
sel -con ainmen , we ecall some use ul p ope ies o he elemen a y closu e.
De ini ion 4.2.3 (Elemen a y closu e). Le Gbe a g oup and le Hbe a subg oup o G. We
de ine he elemen a y closu e o Hin Gas he subg oup
EGpHq“ gPG|dHauspgH, Hq ă 8u.
Lemma 4.2.4. Le H, K be subg oups o Gsuch ha HXKhas ini e index in bo h Hand K,
hen KďEGpHq.
P oo . Le kPKand hPH. Ou goal is o uni o mly bound dpkh, Hq. Since HXKhas ini e
index in H, he e is k0PHXKa uni o mly bounded dis ance om h. No e ha kk0PK. Since
HXKhas ini e index in K, he e is h0PHXKa uni o mly bounded dis ance om kk0. By
iangula inequali y, we ge a uni o m bound on dpkh, h0q.
Recall ha wo g oups H, K a e said o be commensu able i HXKis o ini e index in bo h H
and K. In his chap e we adop a di e en , mo e b oad no ion o commensu abili y.
De ini ion 4.2.5. Le Gbe a g oup and A, B ďGbe subg oups. We say ha Aand Ba e
commensu able i he e exis s gPGsuch ha gAg´1XBhas ini e index in Band AXg´1Bg
has ini e index in A.
Mo eo e , we say ha wo elemen s a, b PGa e non-commensu able i xayand xbya e non-
commensu able in G.
No e ha , in gene al, Hwill no ha e ini e index in EGpHq. A simple example o his is gi en by
conside ing he subg oup xayin xay‘xby – Z2. Indeed, in his case we would ha e EZ2pxayq “ Z2.
This is no he case, howe e , o 2-ended subg oups o hype bolic g oups.
Lemma 4.2.6 ([30, Lemma 6.5]).Le Gbe a hype bolic g oup and Hbe a 2-ended subg oup. Then
EGpHqis 2-ended.
In pa icula , obse e ha EGpHqhas o be he maximal cyclic subg oup con aining H. This
yields he ollowing use ul lemma.
Lemma 4.2.7. Le H1, . . . , Hnbe 2-ended subg oups o a hype bolic g oup G. Then
1. Hiand Hja e commensu able in Gi and only i EGpHiqand EGpHjqa e conjuga e o each
o he .
2. EGpH1q, . . . , EGpHnqu is an almos malno mal collec ion i and only i Hiand Hja e non-
commensu able o e e y i‰j;
4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS111
P oo . Since Hihas ini e index in EGpHiq, we ha e ha EGpHiqand EGpHjqa e commensu able
i and only i Hiand Hja e. In pa icula , his shows one implica ion. Suppose ha EGpHiqand
EGpHjqa e commensu able. Up o conjuga e one o hem we ha e ha gEGpHiqg´1XEGpHjq
has in ini e index in bo h gEGpHiqg´1, and EGpHjq. By Lemma 4.2.4 we ha e gEGpHiqg´1ď
EGpEGpHjqq “ EGpHjqand, by symme y, EGpHjq ď gEGpHiqg´1. Hence, EGpHiqand EGpHjq
a e conjuga e.
Fo he second i em, obse e ha i EGpHiqand EGpHjqa e no commensu able, since hey a e
2-ended g oups i mus ollow |EGpHiqXgEGpHjqg´1|ď8 o all gPG. Hence hey a e almos
malno mal.
We now in oduce he conjugacy g aph associa ed o an edge g oup.
De ini ion 4.2.8 (Commensu abili y class). Le Gbe a g oup and le Pbe a collec ion o 2-
ended subg oups o G. We deno e by « he equi alence ela ion on Pinduced by commensu abili y.
Tha is o say, P1«P2whene e P1, P2a e commensu able (as in De ini ion 4.2.5). Fo each PPP
we use JPK o deno e i s commensu abili y class.
De ini ion 4.2.9 (Equi alence class). Le Gbe a g aph o g oups wi h 2-ended edge g oups.
Conside he mul ise
U“ φe`pGeq, φe´pGeq | ePEpΓqu
o all he images o edge g oups in o e ex g oups coun ed wi h epe i ions.
Le „0be he ela ion on Ude ined by imposing H1„0H2whene e ei he he e exis s esuch
ha H1“φe`pGeqand H2“φe´pGeq, o H1, H2PG o some and H1«H2in G . Ex end
„0 o an equi alence ela ion „on Uby aking he ansi i e closu e o „0.
Fo a e ex g oup H, we deno e by Hsi s equi alence class wi h espec o „.
De ini ion 4.2.10 (Conjugacy g aph). Le Gbe a g aph o g oups wi h 2-ended edge g oups
and le Hsbe he equi alence class o an edge g oup in G. We de ine he conjugacy g aph associa ed
o Hsas he g aph o g oups ∆ Hsde ined as ollows.
Fo each e ex g oup G PG, le Hs “ H1P Hs | H1ďG u.
Ve ices: Fo each e ex o he o iginal g aph Gand commensu abili y class JKKo Hs , add
one e ex K o ∆ Hs. Choose once and o all a ep esen a i e KPJKKand de ine EG pKq o
be he e ex g oup associa ed o K.
Edges: Fo each edge ePΓ such ha φe`pGeq P Hs, add an edge be ween Jφe`pGeqKand
Jφe´pGeqK, wi h associa ed edge g oup Ge. To de ine he edge maps, le Kbe he chosen ep e-
sen a i e o Jφe`pGeqK. Then he e is hPGe`such ha φe`pGeqhĎEGe`pKq. I φe`:GeÑGe`
was he edge map o G, le he a aching map o ∆ Hsbe de ined as φh
e`:GeÑEGe`pKq. No e
ha , by Rema k 4.2.7, his map is well de ined.
112 CHAPTER 4. HYPERBOLIC-2-DECOMPOSABLE GROUPS THAT ARE HHG
Rema k 4.2.11. In his chap e , we conside only g aphs o g oups wi h 2-ended edge g oups. In
pa icula , by Lemma 4.2.6 he e ex g oups o he conjugacy g aphs a e 2-ended. As he edge
g oups o he conjugacy g aphs a e he same as he o iginal edge g oups, he conjugacy g aphs
ha e 2-ended e ex and edge g oups. cons uc ion.
Example 4.2.12. Le F2“ xa, bybe he ee g oup o ank 2 and conside he g oup G o be
π1pGq “ F2˚ a3 ´1“ba2b´1. By cons uc ion, he spli ing o Ghas one e ex wi h associa ed
e ex g oup G “F2and one edge ewi h associa ed cyclic edge g oup Ge. We now cons uc
he conjugacy g aph ∆ Gesassocia ed o Ges. No e i s ha he images o he single edge g oup
a e commensu able in he e ex g oup, as bxa3yb´1Xxba2b´1yis in ini e. Thus, he e is a single
conjugacy class o Gesin F2and, he e o e, a single e ex in ∆ Hs. The associa ed e ex g oup
o ∆ Hsis bEF2pa2qb´1“bxayb´1. The e is also a single edge g oup in ∆ Hswi h associa ed edge
g oup equal o he one in G. The associa ed a aching maps a e φe`and φb
e´. The conjugacy
g aph associa ed o Ges esul s in he g oup xay˚ a2 ´1“a3.
In he ollowing wo lemmas, we desc ibe how is he linea pa ame iza ion in a g aph o 2-ended
g oups ex ended o he gene al se ing using he conjugacy g aph.
Lemma 4.2.13. Le G–π1pGqbe a g aph o hype bolic g oups wi h 2-ended edge subg oups and
le ebe an edge in he unde lying g aph o G. I ∆ Gesdeno es he conjugacy g aph associa ed o
Ges, hen eis unbalanced in Gi and only i π1p∆ Gesqis unbalanced.
P oo . Assume i s ha Gcon ains an unbalanced edge e. The e o e, he e exis s an in ini e
o de elemen aPGeand hPπ1pG´eqsuch ha hφe`paqih´1“φe´paqj o some |i|‰|j|. By
Lemma 1.3.10 he e is a pa h e1, . . . , ekin he g aph o G´ewi h Aep1q“Gα, Bepkq“Gβsuch
ha Bhj
ejXAej`1is non- i ial o e e y j“1, . . . , k ´1 (i.e EGe`
jpBejqhj“EGe`
jpAej`1q) and
elemen s h0PGαand hiPGbpeiqsa is ying
(4.8) p ekhk¨¨¨h1h0qφe`paqip ekhk¨¨¨h1h0q´1“φe´paqj,
o some |i|‰|j|.
This means ha he conjugacy g aph ∆ Gesspli s as π1p∆ Ges´eq˚ e. Recall ha by de ini ion
he a aching maps in ∆ Gesa e de ined as conjuga es φhe1
e1` in Ge1` o he a aching maps φe1` in
G. The e o e, since φe`pgq, φe´pg1qa e conjuga e in π1pGq, ollowing Equa ion (4.8) we ob ain ha
φe`pgqi“φe´pgqjin π1p∆ Ges´eqwhe e |i|‰|j|.
Assume now ha , π1p∆ Gesqis unbalanced. We can apply Lemma 1.3.10 o ob ain,
(4.9) phk k
ek¨¨¨h1 1
e1h0qapphk k
ek¨¨¨h1 1
e1h0q´1“aq,
o some |p| ‰ |q|. He e, ais o in ini e o de , he a ious elemen s hiand abelong o e ex
4.2. HIERARCHICAL HYPERBOLICITY OF HYPERBOLIC-2-DECOMPOSABLE GROUPS113
g oups and a leas one iis non ze o. Ou goal is o modi y he abo e equa ion o ob ain an
analogous one ha holds in π1pGq. Le H0be he e ex g oup o ∆ Ges ha con ains aand le H1
be he o he e ex g oup adjacen o e1in ∆ Ges(possibly, H0“H1). Le xPH1be such ha
p 1
e1h0qapp 1
e1h0q´1“xin π1p∆ Gesq. By de ini ion o conjugacy g aphs, he e a e e ex g oups
G0, G1o Gsuch ha HiďGi. Since he a aching maps in he conjugacy g aph a e de ined as a
conjuga es o he a aching maps o G, he e exis s kiPGisuch ha he ollowing holds in π1pGq:
pk1 1
e1h0k0qappk1 1
e1h0k0q´1“x
Le y1“ pk1 1
e1h0k0q. P oceeding in his way, we ind an elemen yk“yo π1pG´eqsuch ha
yapy´1“aq
wi h |p|‰|q|, showing ha eis unbalanced in G.
Lemma 4.2.14. Le Gbe a g aph o g oups wi h hype bolic e ices and 2-ended edge subg oups.
Suppose, mo eo e , ha o each edge e he conjugacy g aph ∆ Gesis linea ly pa ame izable. Then
π1pGqadmi s a hie a chically hype bolic g oup s uc u e.
P oo . Fo each e ex PVpGqle eiube he se o incoming edges and le EpGe`
iqbe he
elemen a y closu e o he images o he edge g oups in G . Choose ep esen a i es Eiuo he
commensu abili y classes JEpGe`
iqKu. No e ha , by Rema k 4.2.7, Eiu o ms an almos malno -
mal collec ion o subg oups. In pa icula , G is hype bolic ela i e o Eiuby Theo em 4.0.14.
By assump ion, he conjugacy g aph ∆ Gesassocia ed o Gesis linea ly pa ame izable o e e y e.
Tha is o say, o e e y edge e he e exis s Φ Ges:π1p∆ Gesq Ñ Dpeq
8such ha Φ Ges|Gx:GxÑDpeq
8
is a quasi-isome y, whe e Gxis ei he a e ex o edge g oup o ∆ Ges. We endow he a ious
g oups Gxwi h he hie a chical hype bolic s uc u e pGx, Dpeq
8uq as desc ibed in Lemma 4.1.8.
In pa icula , his allows o equip wi h a hie a chically hype bolic g oup s uc u e e e y edge
g oup o Gand e e y g oup EiďG as be o e. No e ha his is well de ined. Indeed, suppose
ha e, a e edges incoming in and Epφe`pGeqq, Epφ `pG qq a e conjuga e. Then e„ and
hence Epφe`pGeqq and Epφ `pG qq a e iden i ied in he conjugacy g aph. Thus he hie a chically
hype bolic s uc u e o he ep esen a i e Edoes no depend on choices. Finally, no e ha since
he i ial hie a chically hype bolic s uc u e on D8sa is y he in e sec ion p ope y and clean
con aine s, so do all he hie a chically hype bolic s uc u es conside ed hus a .
No e ha we a e now in he hypo heses o Theo em 4.2.2, allowing us o equip e e y e ex
g oup wi h a hie a chically hype bolic s uc u e pG ,S q ha u n he edge maps in o glueing
hie omo phisms pGe,SeqãÑ pG ,S q. Mo eo e pG ,S qsa is y he in e sec ion p ope y and
clean con aine s. Applying Theo em 3.3.1 we ob ain ha π1pGqis a hie a chically hype bolic
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122 BIBLIOGRAPHY
Ag adecimien os
Es a secci´on es ´a dedicada a odos aquellos que u ie on un impac o di ec o o indi ec o en es a
esis.
En p ime luga , ag adezco a Mon se a Casals-Ruiz, Ilya Kazachko y Ma k Hagen po la o i-
en aci´on y ayuda que me han b indado du an e odo el cu so de mi o maci´on de doc o ado; po
la gene osidad con la que me han dedicado su iempo y el in aluable apoyo mo al; su ene g´ıa y
dedicaci´on han sido una inspi aci´on pa a m´ı. Es a esis no se hubie a comple ado sin ellos.
G acias a Jason Beh s ock, Ru h Cha ney, Gus a o Fe n´andez-Alcobe , Jon Gonz´alez, Alessand o
Sis o y Gene ie e Walsh po oma se el iempo de lee mi esis y se pa e del ju ado. G a-
cias a Ma hew Du ham y Alexand e Ma in po oma se el iempo de lee y e alua mi esis.
Un ag adecimien o adicional pa a Jason po hospeda mi es ad´ıa en CUNY en e sep iemb e y
diciemb e de 2019.
G acias a mis coau o es, que son excelen es ma em´a icos y de quienes he ap endido y sigo ap en-
diendo.
G acias a FCEyN, UBA po sen a mis bases. Po o ece educaci´on p´ublica, de calidad y g a ui a
a odos y po hace me sen i en casa siemp e que eg eso.
G acias a Na alia, que me ayuda a a a esea los momen os di ´ıciles y es mi cable a ie a en los
buenos; po odo el amo y po elegi nos cada da.
Nunca hubie a es ado donde es oy sin el apoyo incondicional y el amo de mi amilia. Es a esis
ambi´en es ´a dedicada a ellos: a mi mam´a, a mi pap´a, a mis he manos y pad es ex endidos, Ma celo
y Monica.
G acias a And ´es, Fidel y Ma iano po se mis amigos desde chicos; po u u as euniones en VV
despu´es de la plaga.
G acias al inc e´ıble g upo de humanos del depa amen o de ma em´a icas de la UPV: Albe ,
Andoni, Elena, Fede ico, Ike , Ma ialau a, Ma eo, Oihana, Sheila y Xuban. G acias po el ca ´e,
los ompecabezas, los juegos de mesa y po aco a los d´ıas la gos. Un ag adecimien o adicional a
la gen e de BCAM: Dani, Luz y Ja i ˆ2.
Finalmen e, g acias a Euskadi po se un g an luga .
Acknowledgemen s
This sec ion is dedica ed o all o hose who had a di ec o indi ec impac on his hesis.
123
124 BIBLIOGRAPHY
Fi s and o emos , I hank Mon se a Casals-Ruiz, Ilya Kazachko and Ma k Hagen o he
guidance and help ha hey ha e p o ided me h oughou he cou se my PhD aining; o he
gene ous way hey ha e len hei ime o me and he addi ional mo al and uly in aluable
suppo ; hei ene gy and dedica ion has been an inspi a ion o me. This hesis would no ha e
been accomplished wi hou hem.
Thanks o Jason Beh s ock, Ru h Cha ney, Gus a o Fe n´andez-Alcobe , Jon Gonz´alez, Alessand o
Sis o and Gene ie e Walsh o aking he ime o ead my hesis and be pa o he ju y. Thanks
o Ma hew Du ham and Alexand e Ma in o aking he ime o ead and e alua e my hesis.
Addi ional hanks goes o Jason o hos ing my s ay a CUNY be ween Sep embe and Decembe
2019.
Thanks o my coau ho s, who a e ema kable ma hema icians and om whom I ha e lea ned and
con inue o lea n.
Thanks o FCEyN, UBA o se ing my ounda ions. Fo o e ing public, quali y, ee educa ion o
all and o making me eel a home whene e I e u n.
Thanks o Na alia, who manages o keep me going h ough ough imes, and eali y checks me in
good ones; o he all he lo e, and o choosing us e e y day.
I would ha e ne e be whe e I am wi hou he uncondi ional suppo and lo e o my amily. This
hesis is also dedica ed o hem: o my mom, my dad, my b o he s and ex ended pa en s, Ma celo
and Monica.
Thanks o And ´es, Fidel and Ma iano o being my iends since we we e li le; he e’s o hoping
o u u e VV eunions a e he plague has dissipa ed.
Thanks o he amazing g oup o humans in he ma hema ics depa men o he UPV: Albe ,
Andoni, Elena, Fede ico, Ike , Ma ialau a, Ma eo, Oihana, Sheila, and Xuban. Thanks o all he
co ee, he puzzles, he boa dgames and o making long days sho e . Addi ional hanks o he
BCAM people: Dani, Luz, and Ja iˆ2.
Finally, hanks o Euskadi o being an awesome place.