Rigidity, counting and equidistribution of quaternionic Cartan chains
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Rigidi y, coun ing and equidis ibu ion o qua e nionic Ca an chains
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Pa kkonen, Jouni; Paulin, F édé ic
Pa kkonen, J., & Paulin, F. (2022). Rigidi y, coun ing and equidis ibu ion o qua e nionic Ca an
chains. Annales Ma hema iques Blaise Pascal, 28(1), 45-69. h ps://doi.o g/10.5802/ambp.399
2022
ANNALES MATHÉMATIQUES
BLAISE PASCAL
Jouni Pa kkonen & F édé ic Paulin
Rigidi y, coun ing and equidis ibu ion o qua e nionic Ca an chains
Volume 28, no1 (2021), p. 45-69.
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Annales ma héma iques Blaise Pascal 28, 45-69 (2021)
Rigidi y, coun ing and equidis ibu ion
o qua e nionic Ca an chains
Jouni Pa kkonen
F édé ic Paulin
Abs ac
In his pape , we p o e an analog o Ca an’s heo em, saying ha he chain-p ese ing ans o ma ions
o he bounda y o he qua e nionic hype bolic spaces a e p ojec i e ans o ma ions. We gi e a coun ing
and equidis ibu ion esul o he o bi s o a i hme ic chains in he qua e nionic Heisenbe g g oup.
Rigidi é, comp age e équidis ibu ion de chaînes de Ca an qua e nioniennes
Résumé
Dans ce papie , nous mon ons un analogue d’un héo ème de Ca an, disan que les ans o ma ions
du bo d des espaces hype boliques qua e nioniens qui p ése en les chaînes son des ans o ma ions
p ojec i es. Nous donnons un ésul a de comp age e d’équidis ibu ion pou les o bi es de chaînes
a i hmé iques dans le g oupe de Heisenbe g qua e nionien.
1. In oduc ion
The sphe e a in ini y
𝜕∞𝑋
o a nega i ely cu ed symme ic space
𝑋
ca ies many ich
s uc u es, om he geome ic, analy ic and a i hme ic poin s o iew. When he sec ional
cu a u e is no cons an , he possibili ies a e pa icula ly ich, o ins ance wi h he
Ca no –Ca a héodo y, sub-Riemannian o (hype ) CR s uc u es (see o ins ance [
4
,
10
,
12
,
14
,
17
]), leading o s ong igidi y p ope ies, as Pansu’s igidi y heo em o
quasi-isome ies [
18
]. A i hme ic subg oups o he isome y g oup o
𝑋
endow he sphe e
a in ini y o
𝑋
wi h a i hme ic s uc u es, and p oblems o equidis ibu ion o a ional
poin s o sub a ie ies in
𝜕∞𝑋
, as well as in o he homogeneous mani olds, ha e been
in ensi ely s udied (see o ins ance [1, 2, 6, 8, 9, 11, 15, 22] and many o he s).
In his pape , we s udy he qua e nionic hype bolic spaces
𝑋
, whose ex eme igidi y
is exempli ied by he Ma gulis–G omo –Schoen heo em in [
13
], p o ing, con a ily o
he eal o complex case, he a i hme ici y o la ices in he isome y g oup o
𝑋
. As
announced in [
22
], we p o e a on S aud –Ca an ype o igidi y esul o he amily
o all 3-sphe e chains in he sphe e a in ini y o
𝑋
, and, analogously o he complex
hype bolic case ea ed in [
20
], an e ec i e equidis ibu ion esul o he a i hme ic
Keywo ds: coun ing, equidis ibu ion, Ca an chain, qua e nionic Heisenbe g g oup, Cygan dis ance, sub-
Riemannian geome y, qua e nionic hype bolic geome y.
2020 Ma hema ics Subjec Classi ica ion: 11E39, 11F06, 11N45, 20G20, 53C17, 53C55.
45
J. Pa kkonen & F. Paulin
chains in o bi s o a i hme ic g oups buil using maximal o de s in a ional qua e nion
algeb as.
Mo e p ecisely, le
H
be Hamil on’s qua e nion algeb a o e
R
, wi h
𝑥↦→ 𝑥
i s
conjuga ion,
n
:
𝑥↦→ 𝑥𝑥
i s educed no m,
:
𝑥↦→ 𝑥+𝑥
i s educed ace. Le
𝑞
be he
qua e nionic He mi ian o m on he igh ec o space H3o e Hde ined by
𝑞(𝑧0, 𝑧1, 𝑧2)=− (𝑧0𝑧2) +n(𝑧1),
and
PU𝑞
i s p ojec i e uni a y g oup. I is he isome y g oup o he qua e nionic hype bolic
plane H
2
H
, ealised as he nega i e cone o
𝑞
in he igh p ojec i e plane
P2
(H)
, and
no malised o ha e maximal sec ional cu a u e
−
1. See Sec ion 2 o a mo e comple e
desc ip ion.
The bounda y a in ini y
𝜕∞
H
2
H
o H
2
H
is he iso opic cone o
𝑞
in
P2
(H)
, and he
in e sec ions wi h
𝜕∞
H
2
H
o he qua e nionic p ojec i e lines mee ing H
2
H
a e called chains.
We s udy hem, gi ing hei elemen a y p ope ies and comple e geome ic desc ip ions
in Sec ion 3. Ou i s esul is simila o Ca an’s heo em (see [
7
,
10
]) in he complex
hype bolic case. See Theo em 3.3 o a e sion in any dimension.
Theo em 1.1. A chain-p ese ing ans o ma ion om he bounda y a in ini y o he
qua e nionic hype bolic plane o i sel is a p ojec i e uni a y ans o ma ion.
The bounda y a in ini y
𝜕∞
H
2
H
o H
2
H
, wi h he poin
∞=[
1:0:0
]
emo ed,
iden i ies by he map (𝑤0, 𝑤) ↦→ [𝑤0:𝑤: 1]wi h he qua e nionic Heisenbe g g oup
Heis7={(𝑤0, 𝑤) ∈ H×H: 𝑤0=n(𝑤)},
wi h g oup law
(𝑤0, 𝑤)(𝑤0
0, 𝑤0)=(𝑤0+𝑤0
0+𝑤𝑤0, 𝑤 +𝑤0).(1.1)
We endow he me abelian simply connec ed eal Lie g oup
Heis7
wi h i s Cygan dis-
ance
𝑑Cyg
, which is he unique le -in a ian dis ance such ha
𝑑Cyg((𝑤0, 𝑤),(
0
,
0
)) =
(
4
n(𝑤0))1
4
. The chains
𝐶
con ained in
Heis7
a e ellipsoids, and ha e a na u al cen e
cen(𝐶)and adius (see Sec ion 3).
Le
𝐴
be a de ini e (
𝐴⊗QR=H
) qua e nion algeb a o e
Q
, wi h disc iminan
𝐷𝐴
. Le
O
be a maximal o de in
𝐴
. We e e o ins ance o [
25
] o backg ound on qua e nion
algeb as and o de s. The g oup
PU𝑞(O)
o elemen s o
PU𝑞
ep esen ed by ma ices
wi h coe icien s in
O
is a (necessa ily a i hme ic) la ice in
PU𝑞
. A chain
𝐶0
is said o be
a i hme ic o e
O
i he o bi o some poin o
𝐶0
unde he s abilise o
𝐶0
in
PU𝑞(O)
is
dense in
𝐶0
. The s abilise
PU𝑞(O)∞
o
[
1:0:0
]
in
PU𝑞(O)
p ese es he diame e s
o he chains o
𝑑Cyg
. The ollowing esul (see Theo em 4.2 o an explici and mo e
gene al e sion) is an asymp o ic coun ing esul o he a i hme ic chains in an o bi unde
he a i hme ic g oup PU𝑞(O)when hei Cygan diame e ends o 0.
46
Rigidi y, coun ing and equidis ibu ion
Theo em 1.2. Le
𝐶0
be an a i hme ic chain in
𝜕∞
H
2
H
. The e exis s a cons an
𝜅 >
0and
an explici cons an
𝑐 >
0such ha , as
𝜖→
0, he numbe o chains modulo
PU𝑞(O)∞
in he
PU𝑞(O)
-o bi o
𝐶0
, wi h Cygan diame e a leas
𝜖
, is equal o
𝑐𝜖−10(
1
+O(𝜖𝜅))
.
An a i hme ic chain
𝐶0
bounds in H
2
H
a homo he ic copy o he eal hype bolic space o
dimension 4. We deno e by
Co ol(𝐶0)
he olume o he quo ien o his eal hype bolic
space, no malised o ha e sec ional cu a u e
−
1, by he s abilise
PU𝑞(O)𝐶0
o
𝐶0
in
PU𝑞(O)
, and by
𝑚0
he o de o he poin wise s abilise o his eal hype bolic space in
PU𝑞(O)
. We endow he eal Lie g oup
Heis7
wi h i s Haa measu e
Haa Heis7
no malised
in such a way ha he o al mass o he induced measu e on he quo ien o
Heis7
by i s
(uni o m) la ice
Heis7∩(O×O)
is
𝐷2
𝐴
4
(see o ins ance [
22
, Lem. 8
·
4] o an explana ion
o his no malisa ion). Le
𝑚𝐴=
72 i
𝐷𝐴
is e en, and
𝑚𝐴=
1o he wise. Finally, we
deno e by
Δ𝑥
he uni Di ac mass a any poin
𝑥
. The ollowing esul p o es ha he cen e s
o he a i hme ic chains in an o bi unde he a i hme ic g oup
PU𝑞(O)
equidis ibu e in
he qua e nionic Heisenbe g g oup.
Theo em 1.3. Fo he weak-s a con e gence o measu es on Heis7, we ha e
𝑚0𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)
25515 224 Co ol(𝐶0)𝜖10 ∑︁
[𝑔]∈PU𝑞(O)/PU𝑞(O)𝐶0
𝜖≤diam𝑑Cyg (𝑔𝐶0)<∞
Δcen(𝑔𝐶0)
∗
⇀Haa Heis7.
We e e o Sec ion 4 o a e sion wi h cong uences and e o e ms, and a mo e
de elopped s udy o explici examples o a i hme ic chains.
Acknowledgemen s
The au ho s hank he snowy a c ic condi ions in Äkäslompolo in Janua y 2020 which
ha e p o ided an excep ional wo king en i onmen . This esea ch was suppo ed by
CNRS IEA BARP. The second au ho hanks he Labo a oi e de Ma héma iques Jean
Le ay a he Uni e si é de Nan es whe e his pape was comple ed.
2. Qua e nionic hype bolic spaces and Heisenbe g g oups
In his sec ion, we b ie ly ecall some backg ound on he qua e nionic hype bolic
spaces and qua e nionic Heisenbe g g oups, as mos ly con ained in [
22
, §3 and §6], see
also [
16
,
23
] (wi h di e en choices o qua e nionic He mi ian o m and no malisa ion o
he cu a u e).
47
J. Pa kkonen & F. Paulin
Le
H
be Hamil on’s qua e nion algeb a o e
R
, wi h
𝑥↦→ 𝑥
i s conjuga ion,
n
:
𝑥↦→ 𝑥𝑥
i s educed no m,
:
𝑥↦→ 𝑥+𝑥
i s educed ace and
Im
:
𝑥↦→ 1
2(𝑥−𝑥)
i s imagina y
pa map. We deno e by
(
1
, 𝑖, 𝑗, 𝑘)
he canonical basis o
H
as a eal ec o space, so ha
𝑥0+𝑥1𝑖+𝑥2𝑗+𝑥3𝑘=𝑥0−𝑥1𝑖−𝑥2𝑗−𝑥3𝑘. Le
Im H={𝑥∈H: 𝑥=0}=R𝑖+R𝑗+R𝑘
be he
R
-subspace o pu ely imagina y qua e nions o
H
. Fo all
𝑤=(𝑤1, . . . , 𝑤𝑁)
and
𝑤0=(𝑤0
1, . . . , 𝑤0
𝑁)
in he igh ec o space
H𝑁
o e
H
,we deno eby
𝑤·𝑤0=Í𝑁
𝑝=1𝑤𝑝𝑤0
𝑝
hei s anda d qua e nionic He mi ian p oduc , and we de ine
n(𝑤)=𝑤·𝑤=Í𝑁
𝑝=1n(𝑤𝑝)
.
We endow H𝑁wi h he s anda d Euclidean s uc u e (𝑤, 𝑤0) ↦→ 1
2 (𝑤·𝑤0).
We ix
𝑛∈N−{
0
,
1
}
. On he igh ec o space
H×H𝑛−1×H
o e
H
wi h coo dina es
(𝑧0, 𝑧, 𝑧𝑛), le 𝑞be he nondegene a e qua e nionic He mi ian o m
𝑞(𝑧0, 𝑧, 𝑧𝑛)=− (𝑧0𝑧𝑛) +n(𝑧)(2.1)
o Wi signa u e (1, 𝑛), and le Φ:H𝑛+1×H𝑛+1→H, de ined by
Φ:(𝑧0, 𝑧, 𝑧𝑛),(𝑧0
0, 𝑧0, 𝑧0
𝑛)↦→ −𝑧0𝑧0
𝑛−𝑧𝑛𝑧0
0+𝑧·𝑧0,(2.2)
be he associa ed qua e nionic sesquilinea o m.
The Siegel domain model o he qua e nionic hype bolic 𝑛-space H𝑛
His
(𝑤0, 𝑤) ∈ H×H𝑛−1: 𝑤0−n(𝑤)>0,
endowed wi h he Riemannian me ic
d𝑠2
H𝑛
H
=1
( 𝑤0−n(𝑤))2n(d𝑤0−d𝑤·𝑤)+( 𝑤0−n(𝑤))n(d𝑤).
I s bounda y a in ini y is
𝜕∞H𝑛
H=(𝑤0, 𝑤) ∈ H×H𝑛−1: 𝑤0−n(𝑤)=0∪ {∞}.
Aqua e nionic geodesic line in H
𝑛
H
is he image by an isome y o H
𝑛
H
o he in e sec ion
o H
𝑛
H
wi h he qua e nionic line
H× {
0
}
. Wi h ou no malisa ion o he me ic, a
qua e nionic geodesic line is a o ally geodesic submani old o eal dimension 4and
cons an sec ional cu a u e −4.
The closed ho oballs in H𝑛
Hcen ed a ∞ ∈ 𝜕∞H𝑛
Ha e he subse s
H𝑠={(𝑤0, 𝑤) ∈ H𝑛
H: 𝑤0−n(𝑤) ≥ 𝑠},(2.3)
and he ho osphe es cen ed a
∞
a e hei bounda ies
𝜕H𝑠
, whe e
𝑠
anges in
]
0
,+∞[
.
No e ha , o e e y 𝑠∈ ]0,1], we ha e
𝑑(𝜕H1, 𝜕H𝑠)=−ln 𝑠
2.(2.4)
48
Rigidi y, coun ing and equidis ibu ion
The Siegel domain H
𝑛
H
embeds in he igh qua e nionic p ojec i e
𝑛
-space
P𝑛
(H)
by
he map (using homogeneous coo dina es)
(𝑤0, 𝑤) ↦→ [𝑤0:𝑤: 1].
By his map, we iden i y H
𝑛
H
wi h i s image, which when endowed wi h he isome ic
Riemannian me ic, is called he p ojec i e model o H
𝑛
H
. No e ha his image is he
nega i e cone o he qua e nionic He mi ian o m
𝑞
de ined in Equa ion
(2.1)
: we
ha e H
𝑛
H=[𝑧0
:
𝑧
:
𝑧𝑛] ∈ P𝑛
(H)
:
𝑞(𝑧0, 𝑧, 𝑧𝑛)<
0
. This embedding ex ends
con inuously o he bounda y a in ini y, by mapping he poin
(𝑤0, 𝑤) ∈ 𝜕∞
H
𝑛
H−{∞}
o
[𝑤0
:
𝑤
: 1
]
and
∞
o
[
1:0:0
]
, so ha he image o
𝜕∞
H
𝑛
H
is he iso opic cone
o
𝑞
: we ha e
𝜕∞
H
𝑛
H=[𝑧0
:
𝑧
:
𝑧𝑛] ∈ P𝑛
(H)
:
𝑞(𝑧0, 𝑧, 𝑧𝑛)=
0
. A p ojec i e poin
[𝑧0:𝑧:𝑧𝑛] ∈ P𝑛
(H)is posi i e i 𝑞(𝑧0, 𝑧, 𝑧𝑛)>0.
Fo e e y 𝑁∈N, le 𝐼𝑁be he iden i y 𝑁×𝑁ma ix. Le
𝐽=©«
0 0 −1
0𝐼𝑛−10
−1 0 0 ª®®¬
.
The conjuga e- anspose ma ix o a qua e nionic ma ix
𝑋=(𝑥𝑝, 𝑝0)1≤𝑝≤𝑟,1≤𝑝0≤𝑠
in
M𝑟,𝑠 (H)is 𝑋∗=(𝑥∗
𝑝, 𝑝0=𝑥𝑝0, 𝑝)1≤𝑝≤𝑠,1≤𝑝0≤𝑟∈M𝑠,𝑟 (H). Le
U𝑞={𝑔∈GL𝑛+1(H):𝑞◦𝑔=𝑞}={𝑔∈GL𝑛+1(H):𝑔∗𝐽𝑔 =𝐽}
be he uni a y g oup o
𝑞
. I s le linea ac ion on
H𝑛+1
induces a p ojec i e ac ion on
P𝑛
(H)wi h ke nel i s cen e , which is educed o {±𝐼𝑛+1}. The p ojec i e uni a y g oup
PU𝑞=U𝑞/{±𝐼𝑛+1}
o
𝑞
ac s ai h ully on
P𝑛
(H)
, p ese ing H
𝑛
H
, and i s es ic ion o H
𝑛
H
is he ull isome y
g oup o H𝑛
H.
A ma ix
𝑋=©«
𝑎 𝛾∗𝑏
𝛼 𝑀 𝛽
𝑐 𝛿∗𝑑ª®®¬∈GL𝑛+1(H),
49
J. Pa kkonen & F. Paulin
wi h
𝑎, 𝑏, 𝑐, 𝑑 ∈H
,
𝛼, 𝛽, 𝛾, 𝛿 ∈H𝑛−1
(iden i ied wi h hei column ma ices in
M𝑛−1,1(H)
)
and 𝑀∈M𝑛−1,𝑛−1(H), belongs o U𝑞i and only i
𝑐𝑎 −𝛼∗𝛼+𝑎𝑐 =0
𝑑𝑏 −𝛽∗𝛽+𝑏𝑑 =0
−𝛿𝛾∗+𝑀∗𝑀−𝛾𝛿∗=𝐼𝑛−1
𝑑𝑎 −𝛽∗𝛼+𝑏𝑐 =1
𝛿𝑎 −𝑀∗𝛼+𝛾𝑐 =0
𝛿𝑏 −𝑀∗𝛽+𝛾𝑑 =0.
(2.5)
Wi h
Sp(𝑛−
1
)={𝑔∈GL𝑛+1(H)
:
𝑔∗𝑔=𝐼𝑛−1}
, an easy compu a ion shows ha he
block uppe iangula subg oup o U𝑞is
B𝑞=©«
𝜇𝑟 𝜁∗1
2𝑟(n(𝜁) +𝑢)𝜇
0𝑈1
𝑟𝑈𝜁 𝜇
0 0 𝜇
𝑟ª®®¬
:𝜁∈H𝑛−1, 𝑢 ∈Im H,
𝑈∈Sp(𝑛−1), 𝜇 ∈Sp(1), 𝑟 > 0
.
I s image PB𝑞=B𝑞/{±𝐼𝑛+1}in PU𝑞is equal o he s abilise o ∞in PU𝑞.
The qua e nionic Heisenbe g g oup
Heis4𝑛−1
o dimension 4
𝑛−
1is he eal Lie g oup
s uc u e on H𝑛−1×Im Hwi h law
(𝜁, 𝑢)(𝜁0, 𝑢0)=(𝜁+𝜁0, 𝑢 +𝑢0+2 Im 𝜁·𝜁0)
and in e ses
(𝜁, 𝑢)−1=(−𝜁, −𝑢)
. I iden i ies wi h he punc u ed bounda y a in ini y
𝜕∞H𝑛
H−{∞} by he map (𝜁, 𝑢) ↦→ (𝑤0, 𝑤)whe e
(𝑤0, 𝑤)=n(𝜁) +𝑢
2, 𝜁hence (𝜁, 𝑢)=(𝑤, 2 Im 𝑤0),(2.6)
and wi h a subg oup o
PB𝑞⊂PU𝑞
, p ese ing e e y ho oball
H𝑠
o
𝑠 >
0, by he map
(𝜁, 𝑢) ↦→ ±©«
1𝜁∗n(𝜁)+𝑢
2
0𝐼𝑛−1𝜁
0 0 1 ª®®¬
.
Equa ion
(2.6)
allows o eco e he de ini ion o
Heis7
gi en in he In oduc ion, o
which he in e ses a e (𝑤0, 𝑤)−1=(−𝑤0+n(𝑤),−𝑤).
Fo e e y
(𝜁, 𝑢) ∈ Heis4𝑛−1
, he map
(𝜁0, 𝑢0) ↦→ (𝜁, 𝑢)(𝜁0, 𝑢0)
is he Heisenbe g
ansla ion by
(𝜁, 𝑢)
. Fo e e y
𝜁∈H𝑛−1
, he Heisenbe g ansla ion by
(𝜁,
0
)
is
called a ho izon al (Heisenbe g) ansla ion. Fo e e y
𝑢∈Im H
, he Heisenbe g
ansla ion by
(
0
, 𝑢)
is called a e ical (Heisenbe g) ansla ion. The canonical map
Π𝑣
:
Heis4𝑛−1→H𝑛−1
de ined by
(𝜁, 𝑢) ↦→ 𝜁
is a eal Lie g oup mo phism, called he
e ical p ojec ion, whose ke nel is he cen e o
Heis4𝑛−1
. Fo e e y
𝑈∈Sp(𝑛−
1
)
,
50
Rigidi y, coun ing and equidis ibu ion
he map
(𝜁, 𝑢) ↦→ (𝑈𝜁, 𝑢)
is he Heisenbe g o a ion by
𝑈
. Fo e e y
𝜆 >
0, he map
ℎ𝜆:(𝜁, 𝑢) ↦→ (𝜆𝜁, 𝜆2𝑢)is he Heisenbe g dila ion by 𝜆.
The Cygan dis ance
𝑑Cyg
on
Heis4𝑛−1
is he unique le -in a ian dis ance on he eal
Lie g oup Heis4𝑛−1such ha
𝑑Cyg((𝜁, 𝑢),(0,0)) =n(𝜁)2+n(𝑢)1/4,(2.7)
o equi alen ly
𝑑Cyg((𝑤0, 𝑤),(
0
,
0
)) =(
4
n(𝑤0))1
4
by Equa ion
(2.6)
. We in oduce
(see [
19
,
20
] in he complex case) he modi ied Cygan dis ance
𝑑00
Cyg
, as he unique
le -in a ian map om Heis4𝑛−1×Heis4𝑛−1 o [0,+∞[ such ha
𝑑00
Cyg((𝜁, 𝑢),(0,0)) =(n(𝜁)2+n(𝑢))1/2
(n(𝜁)2+n(𝑢))1/2+n(𝜁)1/2,(2.8)
o equi alen ly by Equa ion (2.6)
𝑑00
Cyg((𝑤0, 𝑤),(0,0)) =2n(𝑤0)1/2
(2n(𝑤0)1/2+n(𝑤))1/2.
Though no ac ually a dis ance, he map 𝑑00
Cyg is symme ic and sa is ies
1
√2𝑑Cyg ≤𝑑00
Cyg ≤𝑑Cyg.
Fo e e y nonemp y bounded subse
𝐸
o
Heis4𝑛−1
, we de ine he diame e o
𝐸
o his
almos dis ance as
diam𝑑00
Cyg (𝐸)=sup
𝑥,𝑦 ∈𝐸
𝑑00
Cyg(𝑥, 𝑦).
No e ha he Cygan dis ance and he modi ied Cygan dis ance a e in a ian unde
Heisenbe g ansla ions and o a ions, and ha o e e y
𝜆 >
0, he Heisenbe g dila ion
ℎ𝜆is a homo he y o a io 𝜆 o bo h dis ances.
Lemma 2.1. Fo e e y geodesic line
]𝑥, 𝑦[
in H
𝑛
H
disjoin om he ho oball
H1
, he
dis ance in H𝑛
Hbe ween H1and ]𝑥, 𝑦[is equal o
𝑑(H1,]𝑥, 𝑦[) =−ln 1
√2𝑑00
Cyg(𝑥, 𝑦).
P oo .
By he in a iance unde Heisenbe g ansla ions o
H1
, o he dis ance in H
𝑛
H
and
o he modi ied Cygan dis ance, we may assume ha
𝑥=(𝑤0, 𝑤) ∈ 𝜕∞
H
𝑛
H−{∞,(
0
,
0
)}
and
𝑦=(
0
,
0
) ∈ 𝜕∞
H
𝑛
H− {∞}
. By [
22
, Lem. 6
·
4], he geodesic line om
(𝑤0, 𝑤)
o
(0,0)is, up o ansla ion a he sou ce, he map
𝛾𝑤0,𝑤 :𝑡↦→ 𝑤0(1+𝑒2𝑡𝑤0)−1, 𝑤(1+𝑒2𝑡𝑤0)−1).
51
J. Pa kkonen & F. Paulin
H𝑛−1×Im H
wi h i s eal angen space a
𝑥
, he ibe
𝐸𝑥
o
𝐸
o e
𝑥
is he ho izon al
subspace {(𝜁, 𝑢) ∈ H𝑛−1×Im H:𝑢=0}.
Acalib a ion o
𝐸
is a 1- o m
𝜔
on
𝑊
wi h alues in
Im H
such ha
𝐸=ke 𝜔
. I s
Le i o m is d𝜔. Fo ins ance, in he (𝜁, 𝑢)-coo dina es o Heis4𝑛−1, he o m
𝜔=d𝑢−2 Im(𝜁·d𝜁)
is a calib a ion o
𝐸
(when es ic ed o
𝜕∞
H
𝑛
H−{∞}
). An easy compu a ion shows ha
his calib a ion is in a ian unde Heisenbe g ansla ions and o a ions: Fo e e y such
ans o ma ion
𝛾
, we ha e
𝛾∗𝜔=𝜔
. The ac ha
𝜔
is indeed a calib a ion ollows by
in a iance since
ke
d
𝑢={(𝜁, 𝑢) ∈ H𝑛−1×Im H
:
𝑢=
0
}
. This calib a ion
𝜔
is scaled by
he Heisenbe g dila ions as ollows : o e e y 𝜆 > 0, we ha e (ℎ𝜆)∗𝜔=𝜆2𝜔.
In he ollowing esul , we deno e by
𝑣=𝑣1𝑖+𝑣2𝑗+𝑣3𝑘
he s anda d coo dina e
in
Im H
, and by d
𝑣
he au ological
(Im H)
- alued 1- o m on
Im H
, so ha o e e y
𝑥∈Im H
, he map d
𝑣𝑥
:
𝑇𝑥Im H=Im H→Im H
is he iden i y map. We deno e by
𝜔1, 𝜔2, 𝜔3 he s anda d coo dina es o he calib a ion 𝜔, so ha
𝜔=𝜔1𝑖+𝜔2𝑗+𝜔3𝑘.
Gi en a chain
𝐶
in
𝜕
H
𝑛
H
, le
𝜇=𝜇𝐶
be he (Bo el posi i e) measu e on
Heis4𝑛−1
wi h suppo
𝐶∩Heis4𝑛−1
associa ed wi h he olume o m
𝜔1∧𝜔2∧𝜔3
on
𝐶
. Fo
ins ance, i
𝐶={(𝜁, 𝑢) ∈ H𝑛−1×Im H
:
𝜁=
0
}∪{∞}
is he s anda d e ical chain, hen
𝜔|𝐶=d𝑢|𝐶, so ha 𝜇𝐶is he (in ini e) measu e
𝜇𝐶=d𝑢1d𝑢2d𝑢3,
whose es ic ion o he Euclidean space
𝐶−{∞} ={
0
}×Im H
is he s anda d Lebesgue
measu e.
Gi en a nonze o measu e
𝜇
wi h compac suppo on a ini e dimensional eal a ine
space 𝑉, he ba ycen e (o cen oid) o 𝜇is he poin ba (𝜇)o 𝑉de ined by
ba (𝜇)=1
𝜇(𝑉)∫𝑥∈𝑉
𝑥d𝜇(𝑥).
Fo ins ance, when
𝜇
is suppo ed on a ini e se
𝑆
, hen
ba (𝜇)
is he usual a ine
ba ycen e o he weigh ed amily o poin s 𝑠, 𝜇({𝑠})
𝜇(𝑆)𝑠∈𝑆.
We deno e he open ball o cen e 0and adius
𝑟
in he Euclidean space
Im H
by
𝐵(𝑟)
.
Recall ha he adius o a ini e chain 𝐶is deno ed by 𝑅𝐶.
P oposi ion 3.4. Le 𝐶be a chain in 𝜕∞H𝑛
Hand 𝑐∈𝐶.
(1)
I
𝐶
is a ini e chain, hen he cen e o he chain
𝐶
is equal o he ba ycen e o
he measu e 𝜇𝐶:
cen(𝐶)=ba (𝜇𝐶).
58
Rigidi y, coun ing and equidis ibu ion
(2)
I
𝐶
is a e ical chain, he e is a di eomo phism
𝜏=𝜏𝐶
:
Im H→𝐶−{∞}
such
ha
𝜏∗𝜔=
d
𝑣
, unique up o pos composi ion by a e ical Heisenbe g ansla ion.
Fo e e y Heisenbe g ansla ion o o a ion 𝛾, we ha e 𝜏𝛾𝐶 =𝛾◦𝜏𝐶.
(3)
I
𝐶
is a ini e chain, he e exis s a smoo h di eomo phism
𝜏=𝜏𝐶,𝑐
om
𝐵(
2
𝜋𝑅2
𝐶)
o
𝐶− {𝑐}
, admi ing a con inuous ex ension o
𝜕𝐵(
2
𝜋𝑅2
𝐶)
sending
his sphe e o
𝑐
, such ha
𝜏∗𝜔=
d
𝑣
. This mapping is unique up o pos composi ion
by a Heisenbe g o a ion p ese ing
𝐶
and
𝑐
, and 2
𝜋𝑅2
𝐶
is he unique adius o
which such a mapping exis s.
Fo e e y Heisenbe g ansla ion o o a ion 𝛾, we ha e 𝜏𝛾𝐶,𝛾𝑐 =𝛾◦𝜏𝐶,𝑐.
P oo . (1).
No e ha
Heis4𝑛−1=H𝑛−1×Im H
has a na u al s uc u e o a eal a ine
space, and ha he elemen s o
PB𝑞
ac by a ine ans o ma ions on
Heis4𝑛−1
. This
can be seen o ins ance by saying ha
Heis4𝑛−1
, iden i ied wi h he bounda y o he
p ojec i e model o H
𝑛
H
minus
{∞}
, is a
PB𝑞
-in a ian a ine subspace o he a ine cha
o he qua e nionic p ojec i e space de ined by he qua e nionic p ojec i e hype plane
[𝑧0
:
𝑧
:
𝑧𝑛] ∈ P𝑛
(H)
:
𝑧𝑛=
0
, and ha he qua e nionic p ojec i e ans o ma ions
p ese ing his hype plane ac by a ine ans o ma ions on he associa ed a ine cha .
Ano he way is o check, by an easy compu a ion, ha he Heisenbe g ansla ions,
o a ions and dila ions p ese e he ba ycen e s in he eal a ine space
H𝑛−1×Im H
: Fo
ins ance, o all (𝜁0, 𝑢0),(𝜁, 𝑢),(𝜁0, 𝑢0) ∈ Heis4𝑛−1and 𝑡∈ [0,1], we ha e
(𝜁0, 𝑢0) · 𝑡(𝜁, 𝑢)+(1−𝑡)(𝜁0, 𝑢0)=𝑡(𝜁0, 𝑢0)·(𝜁, 𝑢)+(1−𝑡)(𝜁0, 𝑢0)·(𝜁0, 𝑢0).
In pa icula , he ba ycen e s o measu es
𝜇
wi h compac suppo on
Heis4𝑛−1
a e
equi a ian unde he Heisenbe g ansla ions, o a ions and dila ions: Fo e e y such
ans o ma ion 𝛾, we ha e
ba (𝛾∗𝜇)=𝛾ba (𝜇).(3.3)
In o de o p o e Asse ion
(1)
, by Equa ions
(3.2)
and
(3.3)
, and by he ansi i i y
p ope ies o he Heisenbe g ansla ions and dila ions on chains, we may assume ha
𝑛=
2and ha
𝐶
is a Euclidean sphe e wi h cen e
(
0
,
0
)
and adius 1in he ho izon al
subspace
{(𝜁, 𝑢) ∈ H𝑛−1×H
:
𝑢=
0
}
. Since he
Im H
- alued 1- o m
𝜔|𝐶
is in a ian
unde he Heisenbe g o a ions, he olume o m
𝜔1∧𝜔2∧𝜔3
on
𝐶
is in a ian unde
he Heisenbe g o a ions. Since he only measu e on
𝐶
in a ian unde he Heisenbe g
o a ions is, up o a scala mul iple, he Lebesgue measu e on he Euclidean sphe e
𝐶
, he
measu e
𝜇𝐶
is a mul iple o he Lebesgue measu e on
𝐶
. This can also be p o ed by a
di ec compu a ion: On he Euclidean sphe e 𝐶, wi h 𝜁=𝜁0+𝜁1𝑖+𝜁2𝑗+𝜁3𝑘, we ha e
𝜔1∧𝜔2∧𝜔3=−8
3
∑︁
𝑖=0(−1)𝑖𝜁𝑖d𝜁0∧···∧ c
d𝜁𝑖∧···∧d𝜁3.
59
J. Pa kkonen & F. Paulin
Since he ba ycen e o his measu e is exac ly he o igin
(
0
,
0
)
, which is he cen e o he
ini e chain 𝐶, his p o es Asse ion (1).
(2). Fi s assume ha 𝐶is he s anda d e ical chain
𝐶∞={(𝜁, 𝑢) ∈ H𝑛−1×Im H:𝜁=0} ∪{∞}.
Le
𝜏=𝜏𝐶∞
:
𝑣↦→ (
0
, 𝑣)
. Then
𝜏
is a di eomo phism om
Im H
on o
𝐶∞−{∞}
, such
ha
𝜏∗(
d
𝑢−
2
Im(𝜁
d
𝜁)) =
d
𝑣
. Fo e e y e ical Heisenbe g ansla ion
𝛾
, he map
𝛾◦𝜏
is also a di eomo phism om
Im H
on o
𝐶∞−{∞}
, and since
𝜔
is in a ian unde he
Heisenbe g ansla ions, we also ha e (𝛾◦𝜏)∗𝜔=d𝑣.
I
𝜎
:
Im H→𝐶∞− {∞}
is ano he di eomo phism such ha
𝜎∗𝜔=
d
𝑣
, hen o
e e y
𝑣∈Im H
, we ha e
𝜎0(𝑣)−𝜏0(𝑣) ∈ 𝑇𝐶∞∩ke 𝜔={
0
}
, hus he maps
𝜎
and
𝜏
di e
by an elemen o he ec o subspace
𝐶∞
. The e o e he e exis s a e ical Heisenbe g
ansla ion 𝛾such ha 𝜎=𝛾◦𝜏.
Now, i
𝐶
is ano he e ical chain, he e exis s a composi ion
𝛾
o Heisenbe g ansla-
ions and o a ions such ha
𝐶=𝛾𝐶∞
. De ining
𝜏𝐶=𝛾◦𝜏𝐶∞
gi es a di eomo phism
om
Im H
on o
𝐶−{∞}
such ha
𝜏𝐶∗𝜔=
d
𝑣
, by he in a iance o
𝜔
unde he Heisenbe g
ansla ions and o a ions. This p o es Asse ion (2).
(3). Fi s assume ha 𝐶is he Euclidean 3-sphe e
(𝜁, 𝑢) ∈ H𝑛−1×Im H:n(𝜁1)=𝑅2and 𝑢=𝜁2=··· =𝜁𝑛−1=0,
and ha
𝑐=(𝜁𝑐=(−𝑅,
0
, . . . ,
0
), 𝑢𝑐=
0
)
. No e ha
𝑅
is he adius o he ini e chain
𝐶
.
By he p ope ies o he exponen ial map o he Lie g oup o uni qua e nions, whose
angen space a he iden i y elemen 1is Im H, he smoo h map
𝜏=𝜏𝐶,𝑐 :𝑣↦→ 𝜁=(𝑅𝑒−𝑣/(2𝑅2),0, . . . , 0), 𝑢 =0
om
Im H
o
𝐶
is a di eomo phism om
𝐵(
2
𝜋𝑅2)
on o
𝐶−{𝑐}
. I ex ends con inuously
(and e en smoo hly) o he sphe e
𝜕𝐵(
2
𝜋𝑅2)
, mapping his sphe e o
𝑐
. Conside ing
𝜁
as a unc ion o
𝑣
, we ha e d
𝜁=(− 1
2𝑅𝑒−𝑣/(2𝑅2)
d
𝑣,
0
, . . . ,
0
)
. Hence, since
𝑣
and d
𝑣
a e
pu ely imagina y qua e nions, we ha e
𝜏∗𝜔=−2 Im(𝜁·d𝜁)=−2 Im𝑅𝑒−¯𝑣/(2𝑅2)−1
2𝑅𝑒−𝑣/(2𝑅2)d𝑣=d𝑣.
The uniqueness o
𝜏
up o pos composi ion by a Heisenbe g o a ion p ese ing
𝐶
and
𝑐
,
and he ex ension o he o he chains, ollow as p e iously om he ac ha he chains
a e ans e se o he qua e nionic con ac s uc u e on Heis4𝑛−1and by in a iance o he
calib a ion 𝜔unde he Heisenbe g ansla ions and o a ions.
60
Rigidi y, coun ing and equidis ibu ion
4.
Coun ing and equidis ibu ion o a i hme ic chains in hype sphe ical
geome y
In his sec ion, we p o e (gene alised e sions o ) Theo ems 1.2 and 1.3 o he in oduc ion.
We s a by ecalling a gene al s a emen , coming om a special case o he main esul s
o [21], ha has been made explici in [22].
Le
Γ
be a la ice in
PU𝑞
. Le
𝐷−
and
𝐷+
be nonemp y p ope closed con ex subse s o
H
𝑛
H
, wi h s abilise s
Γ𝐷−
and
Γ𝐷+
in
Γ
espec i ely, such ha he amilies
(𝛾𝐷−)𝛾∈Γ/Γ𝐷−
and
(𝛾𝐷+)𝛾∈Γ/Γ𝐷+
a e locally ini e in H
𝑛
H
. Fo all
𝛾, 𝛾0
in
Γ
, he con ex se s
𝛾𝐷−
and
𝛾0𝐷+
ha e a common pe pendicula i and only i hei closu es
𝛾𝐷−
and
𝛾0𝐷+
in
H
𝑛
H∪𝜕∞
H
𝑛
H
do no in e sec . We deno e by
𝛼𝛾,𝛾0
his common pe pendicula , s a ing
om 𝛾𝐷−a ime 𝑡=0, and by ℓ(𝛼𝛾,𝛾0)i s leng h. The mul iplici y o 𝛼𝛾,𝛾0is
𝑚𝛾,𝛾0=1
ca d(𝛾Γ𝐷−𝛾−1∩𝛾0Γ𝐷+𝛾0−1),
which equals 1 o all
𝛾, 𝛾0∈Γ
when
Γ
ac s eely on
𝑇1
H
𝑛
H
( o ins ance when
Γ
is
o sion- ee). Fo all 𝑠 > 0and 𝑥∈𝜕𝐷−, le
𝑚𝑠(𝑥)=∑︁
𝛾∈Γ/Γ𝐷+:𝐷−∩𝛾𝐷+=∅, 𝛼𝑒,𝛾 (0)=𝑥, ℓ (𝛼𝑒,𝛾)≤𝑠
𝑚𝑒,𝛾
be he mul iplici y o
𝑥
as he o igin o common pe pendicula s wi h leng h a mos
𝑠
om 𝐷− o he elemen s o he Γ-o bi o 𝐷+.
Fo e e y 𝑠 > 0, le
N𝐷−,𝐷+(𝑠)=∑︁
(𝛾,𝛾0)∈Γ ((Γ/Γ𝐷−)×(Γ/Γ𝐷+)) :𝛾𝐷−∩𝛾0𝐷+=∅, ℓ (𝛼𝛾,𝛾0)≤𝑠
𝑚𝛾,𝛾0,
whe e
Γ
ac s diagonally on
Γ×Γ
. When
Γ
has no o sion,
N𝐷−,𝐷+(𝑠)
is he numbe
(wi h mul iplici ies coming om he ac ha
Γ𝐷± 𝐷±
is no assumed o be embedded in
Γ
H
𝑛
H
) o he common pe pendicula s o leng h a mos
𝑠
be ween he images o
𝐷−
and
𝐷+in Γ H𝑛
H.
The ollowing s a emen is a special case o [
22
, Thm. 8
·
1]. We deno e by
Δ𝑥
he uni
Di ac mass a a poin 𝑥.
Theo em 4.1. Le
𝐷−
be a ho oball in H
𝑛
H
cen ed a a pa abolic ixed poin o
Γ
and le
𝐷+
be a qua e nionic geodesic line in H
𝑛
H
such ha
Γ𝐷+ 𝐷+
has ini e olume. Le
𝑚+
be
he o de o he poin wise s abilise o 𝐷+in Γand le
𝑐(𝐷−, 𝐷+)=2(𝑛−1)(2𝑛−1)
𝜋2𝑚+
Vol(Γ𝐷− 𝐷−)Vol(Γ𝐷+ 𝐷+)
Vol(Γ H𝑛
H).
61
J. Pa kkonen & F. Paulin
The e exis s 𝜅 > 0such ha , as 𝑠→ +∞,
N𝐷−,𝐷+(𝑠)=𝑐(𝐷−, 𝐷+)𝑒(4𝑛+2)𝑠1+O(𝑒−𝜅𝑠).
Fu he mo e, he o igins o he common pe pendicula s om
𝐷−
o he images o
𝐷+
unde he elemen s o
Γ
equidis ibu e in
𝜕𝐷−
o he induced Riemannian measu e: As
𝑠→ +∞, we ha e
2(2𝑛+1)Vol(Γ𝐷− 𝐷−)
𝑐(𝐷−, 𝐷+)𝑒−(4𝑛+2)𝑠∑︁
𝑥∈𝜕𝐷−
𝑚𝑠(𝑥)Δ𝑥∗
⇀ ol𝜕𝐷−.(4.1)
Fo smoo h unc ions
𝜓
wi h compac suppo on
𝜕𝐷−
, he e is an e o e m in he
equidis ibu ion claim o Theo em 4.1 when he measu es on bo h sides a e e alua ed on
𝜓
, o he o m
O(𝑒−𝜅𝑠 k𝜓kℓ)
whe e
𝜅 >
0and
k𝜓kℓ
is he Sobole no m o
𝜓
o some
ℓ∈N.
F om now on, we assume ha
𝑛=
2. Le
𝐴
,
𝐷𝐴
,
𝑚𝐴
and
O
be as in he In oduc ion.
We deno e by
|O×|
he o de o he uni g oup o
O
, equal o 24 i
𝐷𝐴=
2, o 12 i
𝐷𝐴=
3, o else o 2,4o 6. See o ins ance [
25
]. As usual, by
Î𝑝|𝐷𝐴
, we mean a p oduc
whe e 𝑝 anges o e he p ime posi i e numbe s di iding 𝐷𝐴.
Fo e e y chain
𝐶
in
𝜕∞
H
2
H
, le
𝐿𝐶
be he qua e nionic p ojec i e line in
P2
(H)
such
ha
𝐶=𝐿𝐶∩𝜕∞
H
2
H
, and le
𝐷𝐶=𝐿𝐶∩
H
2
H
be he associa ed qua e nionic geodesic line.
Fo e e y ini e index subg oup
𝐺
o he a i hme ic la ice
PU𝑞(O)
, we deno e by
𝐺𝐶
he
s abilise o
𝐶
in
𝐺
, by
𝐺∞
he s abilise o
∞
in
𝐺
, and by
Co ol𝐺(𝐶)
he olume o
he o bi old
𝐺𝐶 𝐷𝐶
o he Riemannian me ic o cons an sec ional cu a u e
−
1on
he eal hype bolic 4-space
𝐷𝐶
. Recall ha a chain
𝐶
is a i hme ic o e
O
i and only i
he s abilise in
PU𝑞(O)
(o equi alen ly in
𝐺
) o he qua e nionic geodesic line
𝐷𝐶
has
ini e co olume on 𝐷𝐶.
Theo em 4.2. Le
𝐶0
be an a i hme ic chain o e a maximal o de
O
in a de ini e
qua e nion algeb a o e
Q
. Le
𝐺
be a ini e index subg oup o
PU𝑞(O)
. Then he e exis s
a cons an
𝜅 >
0such ha , as
𝜖 >
0 ends o 0, he numbe
𝜓𝐶0,𝐺 (𝜖)
o chains modulo
𝐺∞in he 𝐺-o bi o 𝐶0wi h 𝑑Cyg-diame e a leas 𝜖is equal o
35 223 36𝐷2
𝐴Co ol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞]
𝜋6𝑚𝐶0,𝐺 𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]𝜖−10 1+O(𝜖𝜅),
whe e 𝑚𝐶0,𝐺 is he o de o he poin wise s abilise o 𝐷𝐶0in 𝐺.
Recall ha he cen e
cen(𝐶)
o a ini e chain
𝐶
is he image o
∞=[
1 : 0 : 0
]
unde
he e lexion on
𝐿𝐶
. The ollowing esul is an equidis ibu ion esul in he qua e nionic
Heisenbe g g oup o he cen e s o he a i hme ic chains in a gi en o bi unde (a ini e
index subg oup o ) PU𝑞(O).
62
Rigidi y, coun ing and equidis ibu ion
Theo em 4.3. Le 𝐶0,𝐺and 𝑚𝐶0,𝐺 be as in Theo em 4.2. As 𝜖 > 0 ends o 0, we ha e
𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]
35 224 36Co ol𝐺(𝐶0)𝜖10 ∑︁
𝐶∈𝐺·𝐶0
diam𝑑Cyg (𝐶)≥𝜖
Δcen(𝐶)
∗
⇀Haa Heis7.
As in Theo em 4.1, he e exis
𝜅 >
0and
ℓ∈N
such ha o e e y smoo h unc ion
𝜓
wi h compac suppo on
Heis7
, he e is an e o e m in his equidis ibu ion esul when
he measu es on bo h sides a e e alua ed on
𝜓
, o he o m
O(𝑠−𝜅k𝜓kℓ)
whe e
k𝜓kℓ
is
he Sobole no m o 𝜓.
We begin by a echnical esul used in he p oo s o he abo e heo ems, which does
no equi e he assump ion
𝑛=
2. Recall ha
𝑑00
Cyg
is he modi ied Cygan dis ance de ined
in Sec ion 2.
Lemma 4.4. Fo e e y 𝑚-chain 𝐶in H𝑛
H, we ha e diam𝑑Cyg (𝐶)=√2 diam𝑑00
Cyg (𝐶).
P oo .
I
𝐶
is a e ical
𝑚
-chain, hen bo h diame e s a e
+∞
. We hence assume ha
𝐶
is
ini e. Since he Heisenbe g ansla ions and o a ions p ese e
𝑑Cyg
and
𝑑00
Cyg
, and by he
ansi i i y p ope ies o he Heisenbe g ansla ions and o a ions on he se o
𝑚
-chains
(see Sec ion 3.2), we may assume ha
𝐶
is a Euclidean sphe e cen e ed a
(
0
,
0
)
wi h
dimension 4
𝑚−
1, con ained in he ho izon al plane
H𝑛−1×{
0
}
o
Heis4𝑛−1
. Since he
Heisenbe g dila ions
(𝜁, 𝑢) ↦→ (𝜆𝜁, 𝜆2𝑢)
wi h
𝜆 >
0a e homo he ies o a io
𝜆
o
𝑑Cyg
and 𝑑00
Cyg, we may assume ha he adius o 𝐶is equal o 1.
Fo e e y
(𝜁,
0
) ∈ 𝐶
, we hus ha e
𝑑Cyg((𝜁,
0
),(
0
,
0
)) =
1by Equa ion
(2.7)
, hence
diam𝑑Cyg (𝐶) ≤ 2by he iangle inequali y. Since
𝑑Cyg((𝜁, 0),(−𝜁, 0)) =𝑑Cyg((𝜁, 0)·(𝜁, 0),(0,0)) =𝑑Cyg((2𝜁, 0),(0,0)) =2,
we ha e diam𝑑Cyg (𝐶)=2.
Using he ansi i i y p ope ies o
Sp(𝑛−
1
)
on he uni sphe e
𝐶
o he Euclidean
space
H𝑛−1
in he same way as in he p oo o [
20
, Lem. 8] in he complex hype bolic
case, we may assume ha 𝑛=3, and ha
diam𝑑00
Cyg (𝐶)=sup
𝑢∈H,𝜙∈[0, 𝜋 ]:n(𝑢)=1
𝑑00
Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0).
63
J. Pa kkonen & F. Paulin
By a compu a ion simila o he one in [
20
, Lem. 8], using Equa ion
(2.8)
and he ac
ha 4n(Im 𝑢)=4− ( 𝑢)2 o any uni qua e nion 𝑢, we ha e
𝑑00
Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0)2
=𝑑00
Cyg (0,0,0),(−1,0,0)·(𝑢cos 𝜙, sin 𝜙, 0)2
=𝑑00
Cyg (0,0,0),𝑢cos 𝜙−1,sin 𝜙, −2 cos 𝜙Im 𝑢)2
=(2−cos 𝜙 𝑢)2+4 cos2𝜙n(Im 𝑢)
((2−cos 𝜙 𝑢)2+4 cos2𝜙n(Im 𝑢))1
2+ (2−cos 𝜙 𝑢)
=2
1
(1+cos2𝜙−cos 𝜙 𝑢)1
2+2− 𝑢cos 𝜙
2(1+cos2𝜙−cos 𝜙 𝑢)
.
As 1+cos2𝜙−cos 𝜙 𝑢≤2−cos 𝜙 𝑢≤4, we ha e
𝑑00
Cyg (1,0,0),(𝑢cos 𝜙, sin 𝜙, 0)2≤2.
Fu he mo e, he equali y holds when 𝑢=1and 𝜙=𝜋. This p o es he esul .
P oo o Theo em 4.2 and Theo em 4.3.
The diame e o a chain o he Cygan dis ance
is in a ian unde he s abilise in
PU𝑞
o he ho osphe e
𝜕H1
, hence is in a ian unde
𝐺∞. The coun ing unc ion 𝜓𝐶0,𝐺 is hus well de ined.
No e ha
H1
is a ho oball cen e ed a he ixed poin o a pa abolic elemen in
PU𝑞(O)
( ake he e ical Heisenbe g ansla ion by
(
0
,
2
𝑢)
o any nonze o
𝑢∈O∩Im H
). We
will apply Theo em 4.1 wi h
Γ = 𝐺
, wi h
𝐷−=H1
, which is hence a ho oball cen e ed a
he ixed poin o a pa abolic elemen in
𝐺
, and wi h
𝐷+=𝐷𝐶0
, which is he qua e nionic
geodesic line in H2
Hwi h bounda y a in ini y equal o 𝐶0. In pa icula 𝑚+=𝑚𝐶0,𝐺.
Le us compu e he cons an
𝑐(𝐷−, 𝐷+)
appea ing in he s a emen o Theo em 4.1.
We ha e Vol(𝐺 H2
H)=[PU𝑞(O):𝐺]Vol(PU𝑞(O) H2
H), whe e, by [22, Thm. 1·4],
Vol(PU𝑞(O) H2
H)=𝜋4𝑚𝐴
175 213 35Ö
𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1),
and by [22, Lem. 8·4],
Vol(Γ𝐷− 𝐷−)=[PU𝑞(O)∞:𝐺∞]Vol(PU𝑞(O)H1 H1)=𝐷2
𝐴[PU𝑞(O)∞:𝐺∞]
160 |O×|2.(4.2)
By de ini ion, we ha e
Vol(Γ𝐷+ 𝐷+)=16 Co ol𝐺(𝐶0),
64
Rigidi y, coun ing and equidis ibu ion
since he sec ional cu a u e o
𝐷+
is cons an
−
4and
𝐷+
has eal dimension 4. We hence
ha e
𝑐(𝐷−, 𝐷+)=35 213 36𝐷2
𝐴Co ol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞]
𝜋6𝑚𝐶0,𝐺𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺].(4.3)
Le
𝑔∈𝐺
be such ha he qua e nionic geodesic line
𝑔𝐷+
is disjoin om
H1
(which
is he case excep o
𝑔
in ini ely many double classes in
𝐺H1 𝐺/𝐺𝐷+
). Le
𝛿𝑔
be he
common pe pendicula om
H1
o
𝑔𝐷+
. I s leng h
ℓ(𝛿𝑔)
is he minimum o he dis ances
om
H1
o a geodesic line be ween wo poin s o
𝜕∞(𝑔𝐷+)=𝑔𝐶0
. Hence, by Lemmas 2.1
and 4.4, we ha e
ℓ(𝛿𝑔)=min
𝑥,𝑦 ∈𝑔𝐶0, 𝑥≠𝑦𝑑(H1,]𝑥, 𝑦[) =−max
𝑥,𝑦 ∈𝑔𝐶0, 𝑥≠𝑦ln 𝑑00
Cyg(𝑥, 𝑦)
√2
=−ln diam𝑑00
Cyg (𝑔𝐶0)
√2=−ln diam𝑑Cyg (𝑔𝐶0)
2.(4.4)
Respec i ely by he de ini ion o he coun ing unc ion
𝜓𝐶0,𝐺
in he s a emen o
Theo em 4.2, since he s abilise o
𝐶0
in
𝐺
is equal o
𝐺𝐷𝐶0=𝐺𝐷+
, by Equa ion
(4.4)
,
by Theo em 4.1, and by Equa ion (4.3), we ha e, as 𝜖 > 0 ends o 0,
𝜓𝐶0,𝐺 (𝜖)
=ca d 𝐺∞ {𝐶∈𝐺·𝐶0: diam𝑑Cyg (𝐶) ≥ 𝜖}
=ca d{[𝑔] ∈ 𝐺∞ 𝐺/𝐺𝐷𝐶0: diam𝑑Cyg (𝑔𝐶0) ≥ 𝜖}
=ca d n[𝑔] ∈ 𝐺H1 𝐺/𝐺𝐷𝐶0:ℓ(𝛿𝑔) ≤ −ln 𝜖
2o+O(1)
=N𝐷−,𝐷+−ln 𝜖
2+O(1)=𝑐(𝐷−, 𝐷+)𝑒−10 ln 𝜖
21+O(𝑒𝜅ln 𝜖
2)
=35 223 36𝐷2
𝐴Co ol𝐺(𝐶0)[PU𝑞(O)∞:𝐺∞]
𝜋6𝑚𝐶0,𝐺𝑚𝐴|O×|2Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]𝜖−10 1+O(𝜖𝜅).
This p o es Theo em 4.2. Le us now p o e Theo em 4.3.
We apply he equidis ibu ion esul in Equa ion
(4.1)
o he o igins
o (𝛿𝑔)
o he
common pe pendicula s
𝛿𝑔
om
𝐷−=H1
o he images
𝑔𝐷+
o
𝑔∈𝐺
. As
𝑠→ +∞
,
we hence ha e, using Equa ions (4.3) and (4.2),
𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]
35 217 36Co ol𝐺(𝐶0)𝑒−10𝑠
∑︁
[𝑔]∈𝐺/𝐺𝐷+:ℓ(𝛿𝑔)≤𝑠
Δo (𝛿𝑔)∗
⇀ ol𝜕H1.(4.5)
65
J. Pa kkonen & F. Paulin
Le
𝑓
:
𝜕∞
H
2
H− {∞} =Heis7→𝜕H1
be he o hogonal p ojec ion map, which is
he homeomo phism
(𝑤0, 𝑤) ↦→ (𝑤0+1
2, 𝑤)
. The push o wa d o he Haa measu e
Haa Heis7by 𝑓is
𝑓∗Haa Heis7=8 ol𝜕H1,(4.6)
see o example he end o he p oo o Theo em 8·3 in [22].
No e ha , o e e y chain
𝐶
, i
𝑟𝐶
is he e lexion on he qua e nionic p ojec i e line
con aining
𝐶
, hen he geodesic line om
∞
o
cen(𝐶)=𝑟𝐶(∞)
, being in a ian unde
𝑟𝐶
, is o hogonal o he qua e nionic geodesic line wi h bounda y a in ini y
𝐶
. Hence o
e e y 𝑔∈𝐺, we ha e
𝑓−1(o (𝛿𝑔)) =cen(𝑔𝐶0).
Le us use in Equa ion
(4.5)
he change o a iables
𝑠=−ln 𝜖
2
and he con inui y o
he push o wa d o measu es by
𝑓−1
. By Equa ions
(4.4)
and
(4.6)
, as
𝜖 >
0 ends o 0,
we ob ain ha he measu es
𝑚𝐶0,𝐺𝑚𝐴𝜋6Î𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)(𝑝3−1)[PU𝑞(O):𝐺]
35 224 36Co ol𝐺(𝐶0)𝜖10 ∑︁
[𝑔]∈𝐺/𝐺𝐷+
diam𝑑Cyg (𝑔𝐶0)≥𝜖
Δcen(𝑔𝐶0)
weak-s a con e ge o he Haa measu e Haa Heis7. This p o es Theo em 4.3.
Example 4.5. Le
𝐶0=[𝑤0
: 0 : 1
] ∈ P2
(H)
:
𝑤0=
0
be he s anda d e ical
chain in
𝜕∞
H
2
H
, which is he in e sec ion o
𝜕∞
H
2
H
wi h he qua e nionic p ojec i e line
𝐿𝐶0=[𝑧0:𝑧1:𝑧2] ∈ P2
(H):𝑧1=0.
An elemen
±©«
𝑎 𝛾∗𝑏
𝛼 𝑀 𝛽
𝑐 𝛿∗𝑑ª®®¬
o
PU𝑞
p ese ing he qua e nionic geodesic line
𝐿𝐶0∩
H
2
H
sa is ies
𝛼𝑤0+𝛽=
0 o
all
𝑤0∈H
wi h
𝑤0>
0. Thus,
𝛼=𝛽=
0, and Equa ions
(2.5)
(o a he he
simila equa ions ob ained by he o mula
𝑋𝑋∗=𝐼𝑛+1
ins ead o
𝑋∗𝑋=𝐼𝑛+1
) imply ha
𝛾=𝛿=
0. Using again Equa ions
(2.5)
, we see ha he s abilise o
𝐿𝐶0
consis s o he
elemen s ©«
𝑎0𝑏
0𝑀0
𝑐0𝑑ª®®¬
such ha (𝑐𝑎)= (𝑑𝑏)=0,𝑐𝑏 +𝑎𝑑 =1and 𝑀∈O×. Thus,
Co olPU𝑞(O)(𝐶0)=𝜋2
1080 Ö
𝑝|𝐷𝐴(𝑝−1)(𝑝2+1)
66
Rigidi y, coun ing and equidis ibu ion
by [5, Thm. 2.5].
The poin wise s abilise o
𝐶0
in
PU𝑞(O)
consis s o he diagonal elemen s wi h
𝑎=𝑑=±1and 𝑀∈O×, gi ing 𝑚𝐶0,PU𝑞(O)=|O×|.
Theo ems 4.2 and 4.3 hen gi e
𝜓𝐶0,PU𝑞(O)(𝜖)=189 220𝐷2
𝐴
𝜋4𝑚𝐴|O×|3Î𝑝|𝐷𝐴(𝑝3−1)𝜖−10 1+O(𝜖𝜅),
and
𝜋4𝑚𝐴|O×|Î𝑝|𝐷𝐴(𝑝3−1)
189 221 𝜖10 ∑︁
𝐶∈PU𝑞(O)·𝐶0: diam𝑑Cyg 𝐶≥𝜖
Δcen(𝐶)∗
⇀Haa Heis7.
Re e ences
[1]
Y es Benois and Hee Oh. E ec i e equidis ibu ion o
𝑆
-in eg al poin s on sym-
me ic a ie ies. Ann. Ins . Fou ie , 62(5):1889–1942, 2012.
[2]
Y es Benois and Jean-F ançois Quin . S a iona y measu es and in a ian subse s o
homogeneous spaces II. J. Am. Ma h. Soc., 26(3):659–734, 2013.
[3] A hu L. Besse. Eins ein mani olds. Classics in Ma hema ics. Sp inge , 2008.
[4]
Oli ie Biqua d. Qua e nionic con ac s uc u es. In P oceedings o he Second
Mee ing on Qua e nionic S uc u es in Ma hema ics and Physics (Roma, 1999),
pages 23–30. Wo ld Scien i ic, 2001.
[5]
S e an B eulmann and Volke Helmke. The co olume o qua e nion g oups on he
ou -dimensional hype bolic space. Ac a A i h., 77(1):9–21, 1996.
[6]
Anne B oise-Alamichel, Jouni Pa kkonen, and F édé ic Paulin. Equidis ibu ion
and coun ing unde equilib ium s a es in nega i e cu a u e and ees. Applica-
ions o non-A chimedean Diophan ine app oxima ion, olume 329 o P og ess in
Ma hema ics. Bi khäuse , 2019.
[7]
Élie Ca an. Su le g oupe de la géomé ie hype sphé ique. Commen . Ma h. Hel .,
4:158–171, 1932.
[8]
William Duke. Ra ional poin s on he sphe e. Ramanujan J., 7(1-3):235–239, 2003.
[9]
Jo dan S. Ellenbe g, Philippe Michel, and Akshay Venka esh. Linnik’s e godic
me hod and he dis ibu ion o in ege poin s on sphe es. In Au omo phic ep esen a-
ions and L- unc ions, olume 22 o Ta a Ins i u e o Fundamen al Resea ch S udies
in Ma hema ics, pages 119–185. Ta a Ins i u e o Fundamen al Resea ch, 2013.
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