L ’Enseignement Mathématiq ue (2) 63 (2017), 333– 373 DOI 10.4171/LEM/63-3/4-5
The h yperbolic g eome try of Mar k o v’ s theorem on
Diophantine appr o ximation and quadratic f orms
Bor is Sprin gborn
A bstract. Mark o v’ s theorem classifies the w ors t ir rational numbers with respect to rational
appro ximation and the indefinite binar y quadratic f or ms whose values f or integer ar guments
sta y f ar thest a w a y from zero. The main purpose of this paper is to present a ne w proof
of Mark o v’ s theorem using h yperbolic g eometry . The main ingredients are a dictionar y to
translate betw een h yperbolic geometry and alg ebra/number theor y , and some v er y basic
tools bor ro wed from modern g eometr ic T eichmüller theory . Simple closed g eodesics and
ideal tr iangulations of the modular torus play an important role, and so do the problems:
Ho w far can a s traight line crossing a triangle sta y aw a y from the v er tices? Ho w f ar can
it sta y a w a y from all v er tices of the tessellation g enerated b y this tr iangle? Definite binar y
quadratic f or ms are briefly discussed in the last section.
Mathematics Subject Classification (2010). Pr imary: 11J06, 32G15.
K e yw ords. Modular torus, simple closed geodesic, Mark o v eq uation, F ord circles, F arey
tessellation.
1. Introduction
The main pur pose of this ar ticle is to present a ne w proof of Mark o v’ s
theorem [ 48 , 49 ] (Secs. 2 , 3 ) using h yperbolic g eometr y . R oughl y , the dictionar y
sho wn on the f ollo wing pag e is used to translate betw een h yperbolic g eometr y
and alg ebra/number theory .
The proof is based on P enner ’ s g eometr ic inter pretation of Mark o v’ s equa-
tion [ 55 , p. 335 f] (Sec. 12 ), and the main tools are bor ro w ed from his theor y of
decorated T eic hmüller space (Sec. 11 ). Ultimatel y , the proof of Mark o v’ s theorem
boils do wn to the question:
Ho w far can a s traight line crossing a tr iangle sta y a w a y from all
v er tices?

334 B. Springb orn
Dictionar y : Hyperbolic Geometry – Algebra/N umber Theor y
Hyperbolic Geometr y Alg ebra/N umber Theory
horocy cle nonzero v ector .p ; q / 2 R 2 Sec. 5
g eodesic indefinite binar y quadratic f or m f Sec. 10
point definite binar y quadratic f or m f Sec. 16
signed distance betw een horocy cles 2 log ˇ ˇ ˇ det  p 1 p 2
q 1 q 2  ˇ ˇ ˇ ( 24 )
signed distance betw een horocy cle and
g eodesic/point
log f .p ; q /
p j det f j ( 29 ) ( 46 )
ideal tr iangulation of the modular torus Mark o v tr iple Sec. 12
It is fun and a recommended e x ercise to consider this q uestion in elementary
euclidean g eometry . Here, w e need to deal with ideal hyperbolic triangles,
decorated with horocy cles at the v er tices, and “dis tance from the v er tices ” is
to be understood as “signed dis tance from the horocy cles ” (Sec. 13 ).
The subjects of this ar ticle, Diophantine appro ximation, quadratic f or ms,
and the h yperbolic geometry of numbers, are connected with div erse areas
of mathematics and its applications, ranging from from the ph y llotaxis of
plants [ 16 ] to the stability of the solar sy stem [ 37 ], and from Gauss ’ Disquisitiones
Arithme ticae to Mirzakhani’ s Fields Medal [ 53 ]. An adeq uate sur v e y of this area,
e v en if limited to the most impor tant and most recent contributions, w ould be
be y ond the scope of this introduction. The books b y Aigner [ 2 ] and Cassels [ 11 ] are
e x cellent ref erences f or Mark o v’ s theorem, Bombier i [ 6 ] pro vides a concise proof,
and more about the Mark o v and Lagrang e spectra can be f ound in Mal y shev’ s
sur v e y [ 47 ] and the book b y Cusick and Flahiv e [ 20 ]. The f ollo wing discussion
f ocuses on a f e w historic sources and the most immediate conte xt and is f ar from
comprehensiv e.
One can distinguish tw o approaches to a g eometric treatment of continued
fractions, Diophantine appro ximation, and quadratic f or ms. In both cases, number
theor y is connected to g eometry b y a common symmetr y group, GL 2 . Z / . The first
approach, kno wn as the geometry of numbers and connected with the name of
Mink o w ski, deals with the g eometr y of the Z 2 -lattice. Klein inter preted continued
fraction appro ximation, intuitiv el y speaking, as “pulling a thread tight ” around
lattice points [ 41 , 42 ]. This approach e xtends naturally to higher dimensions,
leading to a multidimensional g eneralization of continued fractions that w as
championed b y Ar nold [ 3 , 4 ]. Delone ’ s comments on Mark o v’ s w ork [ 22 ] also
belong in this categor y (see also [ 29 ]).

The h yperbolic g eometr y of Mark o v’ s theorem 335
In this ar ticle, w e pursue the other approac h in v ol ving Ford circles and the
F are y tessellation of the h yperbolic plane (Fig. 6 ). This approac h could be called
the h yperbolic geometry of numbers. Bef ore Ford’ s g eometr ic proof [ 27 ] of
Hur witz’ s theorem [ 38 ] (Sec. 2 ), Speiser had apparentl y used the F ord circles to
pro v e a w eaker appro ximation theorem. Ho w ev er , onl y the f ollo wing note sur viv es
of his talk [ 70 , m y translation]:
A g eometric figur e relat ed to number theor y . If one cons tr ucts in the
upper half plane f or e v er y rational point of the x -axis with abscissa p
q
the circle of radius 1
2 q 2 that touc hes this point, then these circles do
not o v er lap an ywhere, onl y tang encies occur . The domains that are not
co v ered consis t of circular triangles. Follo wing the line x D ! (ir rational
number) do wn w ard to w ards the x -axis, one intersects infinitel y man y
circles, i.e., the inequality
ˇ ˇ !  p
q ˇ ˇ < 1
2 q 2
has infinitel y man y solutions. The y cons titute the appro ximations b y
Mink o wski’ s continued fractions.
If one increases the radii to 1
p 3q 2 , then the gaps close and one obtains
the theorem on the maximum of positiv e binar y quadratic f or ms.
See R em. 9.2 and Sec. 16 f or br ief comments on these theorems. Based
on Speiser ’ s talk, Züllig [ 75 ] dev eloped a comprehensiv e g eometr ic theor y of
continued fractions, including a g eometric proof of Hur witz’ s theorem.
Both Züllig and F ord treat the ar rang ement of F ord circles using elementary
euclidean g eometry and do not mention an y connection with h yperbolic geometry .
In Sec. 9 , w e transf er their proof of Hur witz’ s theorem to h yperbolic g eometry .
The conceptual adv antag e is obvious: One has to consider onl y three circles
instead of infinitel y man y , because all tr iples of pair wise touching horocy cles are
congr uent.
T oda y , the role of h yperbolic geometry is w ell unders tood. Continued fraction
e xpansions encode directions f or navig ating the F are y tessellation of the h yperbolic
plane [ 7 , 33 , 67 ]. In f act, muc h w as already kno wn to Hur witz [ 39 ] and
Klein [ 40 , 42 ]. A ccording to Klein [ 42 , p. 248], the y built on Her mite ’ s [ 35 ]
purel y alg ebraic disco v er y of an in v ar iant “incidence ” relation betw een definite
and indefinite f or ms, which the y translated into the languag e of g eometr y . While
Hur witz and Klein ne v er mention horocy cles, the y kne w the other entr ies of
the dictionar y , and ev en use the F are y tr iangulation. In the Ca y le y–Klein model
of h yperbolic space, the g eometr ic inter pretation of binar y quadratic f or ms is
easil y es tablished: The projectivized v ector space of real binar y quadratic f or ms

336 B. Springb orn
is a real projectiv e plane and the deg enerate f or ms are a conic section. Definite
f or ms cor respond to points inside this conic, hence to points of the h yperbolic
plane, while indefinite f or ms cor respond to points outside, hence, b y polarity , to
h yperbolic lines. From this geometric point of vie w , Klein and Hur witz discuss
classical topics of number theor y lik e the reduction of binary quadratic f or ms,
their automor phisms, and the role of P ell’ s equation. Strang el y , it seems they
ne v er treated Diophantine appro ximation or Mark o v’ s w ork this w a y .
Cohn [ 12 ] noticed that Mark o v’ s Diophantine equation ( 4 ) can easil y be
obtained from an elementar y identity of Fr ic k e in v ol ving the traces of 2 
2 -matr ices. Based on this alg ebraic coincidence, he de v eloped a g eometr ic
inter pretation of Mark o v f or ms as simple closed g eodesics in the modular
tor us [ 13 , 14 ], whic h is also adopted in this ar ticle.
A muc h more g eometric inter pretation of Mark o v’ s equation w as disco v ered
b y Penner (as mentioned abo v e), as a b yproduct of his decorated T eic hmüller
theor y [ 55 , 56 ]. This inter pretation f ocuses on ideal tr iangulations of the modular
tor us, decorated with a horocy cle at the cusp, and the w eights of their edg es
(Sec. 12 ). P enner ’ s inter pretation also e xplains the role of simple closed g eodesics
(Sec. 14 ).
Mark o v’ s or iginal proof (see [ 6 ] f or a concise moder n e xposition) is based on
an anal y sis of continued fraction e xpansions. Using the inter pretation of continued
fractions as directions in the F are y tessellation mentioned abo v e, one can translate
Mark o v’ s proof into the language of h yperbolic g eometr y . The anal y sis of allo w ed
and disallo w ed subsequences in an e xpansion translates to symbolic dynamics of
g eodesics [ 66 ].
In his 1953 thesis, whic h w as published muc h later , Gorshk o v [ 30 ] pro vided a
g enuinel y ne w proof of Mark o v’ s theorem using h yperbolic g eometry . It is based
on tw o impor tant ideas that are also the f oundation f or the proof presented here.
Firs t, Gorshko v realized that one should consider all ideal tr iangulations of the
modular tor us, not onl y the projected F are y tessellation. This reduces the symbolic
dynamics argument to almos t nothing (in this ar ticle, see Proposition 15.1 , the
proof of implication “ (c) ) (a) ”). Second, he understood that Mark o v’ s theorem
is about the distance of a g eodesic to the v ertices of a tr iangulation. Ho w ev er ,
lac king moder n g eometr ic tools of T eichmüller theory (like horocy cles), Gorshk o v
w as not able to treat the g eometr y of ideal tr iangulations directly . Ins tead, he
considers compact tor i composed of tw o equilateral h yperbolic tr iangles and lets
the side length tend to infinity . The compact tor i ha v e a cone-lik e singularity at
the v er te x, and the de v eloping map from the punctured tor us to the h yperbolic
plane has infinitel y man y sheets. This limiting process complicates the argument
considerabl y . Also, the tr igonometr y becomes simpler when one needs to consider
onl y decorated ideal triangles. Gorshk o v’ s decision “not to restrict the e xposition to

The h yperbolic g eometr y of Mark o v’ s theorem 337
the minimum necessar y f or pro ving Marko v’ s theorem but rather to e x ecute it with
considerable completeness, retaining e v er ything that is of independent interest ”
mak es it harder to recognize the main lines of ar gument. This, tog ether with an
undul y dismissiv e MathSciNet re vie w , ma y account f or the lac k of recognition
his w ork receiv ed.
In this ar ticle, w e adopt the opposite s trategy and s tic k to pro ving Mark o v’ s
theorem. Man y natural generalizations and related topics are be y ond the scope of
this paper , f or e x ample the appro ximation of comple x numbers [ 21 , 25 , 26 , 61 ],
g eneralizations to other Riemann surfaces or discrete groups [ 1 , 5 , 9 , 31 , 46 , 62 , 63 ],
higher dimensional manif olds [ 36 , 73 ], other Diophantine appro ximation theorems,
f or e xample Khinchin ’ s [ 71 ], and the asymptotic g ro wth of Mark o v numbers and
lengths of closed g eodesics [ 8 , 50 , 52 , 68 , 69 , 7 4 ]. Is the treatment of Mark o v’ s
equation using 3  3 -matr ices [ 57 , 59 ] related? Do the methods presented here
help to co v er a lar g er par t of the Mark o v and Lag rang e spectra b y consider ing
more complicated g eodesics [ 18 , 17 , 19 ]? Can one treat, sa y , ter nar y quadratic
f or ms or binar y cubic f or ms in a similar f ashion?
The notor ious U niqueness Conjecture f or Mark o v numbers (R em. 2.1 (iv)),
which goes bac k to a neutral statement b y Frobenius [ 28 , p. 461], sa y s in g eometr ic
ter ms: If tw o simple closed geodesics in the modular torus ha v e the same length,
then the y are related b y an isometr y of the modular tor us [ 65 ]. Eq uiv alentl y , if tw o
ideal arcs ha v e the same w eight, the y are related this w a y . Hyperbolic geometry
w as ins trumental in pro ving the uniqueness conjecture f or Mark o v numbers that
are pr ime po w ers [ 10 , 44 , 64 ]. W ill g eometry also help to settle the full U niqueness
Conjecture, or is it “a conjecture in pure number theor y and not tractable b y
h yperbolic geometry arguments ” [ 51 ]? Will combinator ial methods succeed? Who
kno ws. These ma y not ev en be v er y meaningful ques tions, like asking: “W ill a
proof be easier in English, French, R ussian, or Ger man?” On the other hand,
sometimes it helps to speak more than one languag e.
2. The w orst irrational numbers
There are tw o v ersions of Mark o v’ s theorem. One deals with Diophantine
appro ximation, the other with quadratic f or ms. In this section, w e recall some
related theorems and s tate the Diophantine appro ximation v ersion in the f or m in
which w e will pro v e it (Sec. 15 ). The f ollo wing section is about the q uadratic
f or ms v ersion.
Let x be an ir rational number . For e v ery positiv e integ er q there is ob viousl y
a fraction p
q that appro ximates x with error less than 1
2 q . If one chooses

338 B. Springb orn
denominators more carefull y , one can find a sequence of fractions con v er ging to
x with er ror bounded b y 1
q 2 :
Theorem. F or ev er y irrational number x , ther e ar e infinit ely many fr actions p
q
satisfying
ˇ ˇ ˇ x  p
q ˇ ˇ ˇ < 1
q 2 :
This theorem is sometimes attr ibuted to Dir ic hlet although the s tatement had
“long been kno wn from the theor y of continued fractions ” [ 23 ]. In fact, Dirichlet
pro vided a par ticularl y simple proof of a multidimensional generalization, using
what later became kno wn as the pigeonhole principle.
Klaus R oth w as a warded a F ields Medal in 1958 f or sho wing that the e xponent 2
in Dir ic hlet ’ s appro ximation theorem is optimal [ 60 ]:
Theorem (R oth). Suppose x and ˛ ar e real number s, ˛ > 2 . If t her e ar e infinitely
many r educed fractions p
q satisfying
ˇ ˇ ˇ x  p
q ˇ ˇ ˇ < 1
q ˛ ;
then x is tr anscendental.
In other w ords, if the e xponent in the er ror bound is greater than 2 then
alg ebraic ir rational numbers cannot be appro ximated. This is an e x ample of a
g eneral observation: “From the point of vie w of rational appro ximation, the
simplest number s ar e the w orst ” (Hardy & W r ight [ 32 ], p. 209, their emphasis).
R oth ’ s theorem sho w s that the w ors t ir rational numbers are alg ebraic. Mark o v’ s
theorem, which w e will state shortly , sho ws that the w orst alg ebraic ir rationals
are quadratic.
While the e xponent is optimal, the constant f actor in Dir ichlet ’ s appro ximation
theorem can be impro v ed. Hurwitz [ 38 ] sho w ed that the optimal constant is 1
p 5 ,
and that the golden ratio belongs to the class of v er y w ors t ir rational numbers:
Theorem (Hurwitz). (i) F or ev er y irr ational number x , ther e ar e infinit ely many
fr actions p
q satisfying
(1) ˇ ˇ ˇ x  p
q ˇ ˇ ˇ < 1
p 5 q 2 :
(ii) If  > p 5 , and if x is equiv alent t o the golden r atio  D 1
2 .1 C p 5/ , then
ther e ar e only finit ely many fr actions p
q satisfying
(2) ˇ ˇ ˇ x  p
q ˇ ˇ ˇ < 1
 q 2 :

The h yperbolic g eometr y of Mark o v’ s theorem 339
T w o real numbers x , x 0 are called equiv alent if
(3) x 0 D a x C b
c x C d ;
f or some integ ers a , b , c , d satisfying
j a d  b c j D 1:
If infinitel y man y fractions satisfy ( 2 ) f or some x , then the same is true f or any
equiv alent number x 0 . This f ollo w s simply from the identity
.q 0 / 2 ˇ ˇ ˇ x 0  p 0
q 0 ˇ ˇ ˇ D q 2 ˇ ˇ ˇ x  p
q ˇ ˇ ˇ ˇ ˇ c  p
q  C d ˇ ˇ
ˇ ˇ c x C d ˇ ˇ
;
where x and x 0 are related b y ( 3 ) and p 0 D a p C b q , q 0 D c p C d q . (N ote that
the last f actor on the r ight hand side tends to 1 as p
q tends to x .)
Hur witz also states the f ollo wing results, “whose proofs can easil y be obtained
from Mark o v’ s in v estig ation ” of indefinite quadratic f or ms:
 If x is an ir rational number not eq uiv alent to the golden ratio  , then
infinitel y man y fractions satisfy ( 2 ) with  D 2 p 2 .
 F or an y <3 , there are onl y finitel y man y equiv alence classes of numbers
that cannot be appro ximated, i.e., f or which there are onl y finitely man y
fractions satisfying ( 2 ) . But f or  D 3 , there are infinitel y man y classes that
cannot be appro ximated.
Hur witz stops here, but the s tory continues. T able 1 lists representativ es x of
the fiv e w ors t classes of ir rational numbers, and the larg est v alues L.x / f or 
f or which there e xist infinitel y man y fractions satisfying ( 2 ) . For e xample, p 2
belongs to the class of second w orst ir rational numbers. The las t tw o columns
will be e xplained in the statement of Mark o v’ s theorem.
Mark o v’ s theorem establishes an e xplicit bi jection betw een the equiv alence
classes of the w orst ir rational numbers, and sor ted Mark o v tr iples. Here, w ors t
irr ational number s means precisel y those that cannot be appro ximated f or some
<3 . A Marko v triple is a triple .a ; b ; c / of positiv e integers satisfying Mark o v’ s
equation
(4) a 2 C b 2 C c 2 D 3a b c :
A Marko v number is a number that appears in some Mark o v triple. An y
per mutation of a Mark o v tr iple is also a Mark o v triple. A sor ted Mar ko v triple
is a Mark o v tr iple .a; b ; c / with a  b  c .
W e revie w some basic f acts about Mark o v tr iples and ref er to the literature f or
details, f or e xample [ 2 , 11 ]. Firs t and f oremost, note that Mark o v’ s equation ( 4 ) is

340 B. Springb orn
T able 1
The fiv e w orst classes of irrational numbers
Rank x L.x / a b c p 1 p 2
1 1
2 .1 C p 5/ p 5 D 2 :2 : : : 1 1 1 0 1
2 p 2 2 p 2 D 2 :8 : : : 1 1 2  1 1
3 1
10 .9 C p 221/ 1
5 p 221 D 2 :97 : : : 1 2 5  1 2
4 1
26 .23 C p 1517/ 1
13 p 1517 D 2 :996 : : : 1 5 13  3 2
5 1
58 .5 C p 7565/ 1
29 p 7565 D 2 :9992 : : : 2 5 29  7 3
quadratic in eac h v ar iable. This allo w s one to generate ne w solutions from kno wn
ones: If .a; b ; c / is a Marko v tr iple, then so are its neighbors
(5) .a 0 ; b ; c /; .a ; b 0 ; c /; .a; b ; c 0 /;
where
(6) a 0 D 3b c  a D b 2 C c 2
a ;
and similar l y f or b 0 and c 0 . Hence, there are three in v olutions  k on the set of
Mark o v tr iples that map an y tr iple .a ; b ; c / to its neighbors:
(7)  1 .a ; b ; c / D .a 0 ; b ; c /;  2 .a ; b ; c / D .a ; b 0 ; c /;  3 .a ; b ; c / D .a ; b ; c 0 /:
These in v olutions act without fix ed points and e v er y Mark o v tr iple can be obtained
from a single Mark o v tr iple, f or e xample from .1; 1; 1/ , b y appl ying a composition
of these in v olutions. The sequence of in v olutions is uniquel y determined if one
demands that no tr iple is visited twice. Thus, the solutions of Mark o v’ s equation ( 4 )
f or m a tr iv alent tree, called the Mar ko v tr ee , with Mark o v tr iples as v er tices and
edg es connecting neighbors (see F ig. 1 ).
Theorem (Mark o v , Diophantine appro ximation v ersion). (i) Let .a ; b ; c / be any
Marko v triple, let p 1 , p 2 be int eg ers satisfying
(8) p 2 b  p 1 a D c ;
and let
(9) x D p 2
a C b
a c  3
2 C r 9
4  1
c 2 :

The h yperbolic g eometr y of Mark o v’ s theorem 341
1
1
1
2
2
2
5
5
5
5
5
5
29
29
29
29
29
29
13
13
13
13
13
13
169
169
169 169
169
169
433
433
433
433
433
433
194
194
194
194
194
194
34
34
34
34
34
34
a b
c a 0
b
c
a b
c 0
a c
b 0
Figure 1
Mark o v tree
Then ther e ar e infinit ely many fr actions p
q satisfying ( 2 ) with
(10)  D r 9  4
c 2 ;
but only finitely many f or any lar g er value of  .
(ii) Conv ersely , suppose x 0 is an irr ational number suc h that only finit ely many
fr actions p
q satisfy ( 2 ) f or some <3 . Then ther e exis ts a unique sorted Mar kov
triple .a ; b ; c / suc h that x 0 is equiv alent to x defined by eq uation ( 9 ) .
R emar k 2.1. A f e w remarks, firs t some ter minology .
(i) The Lag r ang e number L.x / of an ir rational number x is defined b y
L.x / D sup ®  2 R ˇ ˇ infinitel y man y fractions p
q satisfy ( 2 ) ¯ ;
and the set of Lagrang e numbers ¹ L.x / j x 2 R n Q º is called the Lagr ang e
spectrum . Equation ( 10 ) descr ibes the par t of the Lagrange spectrum belo w 3 ,
and equation ( 9 ) pro vides representativ es of the cor responding equiv alence classes
of ir rational numbers.
(ii) It ma y seem strang ely unsymmetric that p 2 appears in equation ( 9 ) and
p 1 does not. The appearance is deceptiv e: Mark o v’ s equation ( 4 ) and equation ( 8 )
impl y that eq uation ( 9 ) is equiv alent to
x D p 1
b  a
b c C 3
2 C r 9
4  1
c 2 :

342 B. Springb orn
(iii) The three integ ers of a Mark o v triple are pair wise copr ime. (This is tr ue
f or .1; 1; 1/ , and if it is tr ue f or some Mark o v tr iple, then also f or its neighbors.)
Theref ore, integ ers p 1 , p 2 satisfying ( 8 ) alwa y s e xist. Different solutions .p 1 ; p 2 /
f or the same Mark o v tr iple lead to equiv alent values of x , differ ing b y integ ers.
(iv) The f ollo wing question is more subtle: U nder what conditions do differ ent
Mark o v tr iples .a ; b ; c / and .a 0 ; b 0 ; c 0 / lead to eq uiv alent numbers x , x 0 ? Clear l y ,
if c 6D c 0 , then x and x 0 are not equiv alent because  6D  0 . But Mark o v tr iples
.a ; b ; c / and .b ; a; c / lead to equiv alent numbers. In g eneral, the numbers x
obtained b y ( 9 ) from Marko v tr iples .a; b ; c / and .a 0 ; b 0 ; c 0 / are equiv alent if
and onl y if one can g et from .a ; b ; c / to .a 0 ; b 0 ; c 0 / or .b 0 ; a 0 ; c 0 / b y a finite
composition of the in v olutions  1 and  2 fixing c . In this case, let us consider
the Mark o v tr iples equiv alent . Ev ery equiv alence class of Mark o v tr iples contains
e xactl y one sor ted Mark o v triple. It is not kno wn whether there e xists onl y one
sor ted Mark o v tr iple .a ; b ; c / f or e v er y Mark o v number c . This w as remarked
b y Frobenius [ 28 ] some one hundred y ears ago, and the ques tion is s till open.
The affir mativ e statement is kno wn as the U niq ueness Conjectur e for Mar kov
N umber s . Consequentl y , it is not kno wn whether there is only one eq uiv alence
class of numbers x f or ev er y Lagrang e number L.x / < 3 .
(v) The attr ibution of Hur witz’ s theorem ma y seem s trang e. It co v ers only
the simplest part of Mark o v’ s theorem, and Mark o v’ s w ork precedes Hur witz’ s.
Ho w e v er , Mark o v’ s or iginal theorem dealt with indefinite quadratic f or ms (see the
f ollo wing section). Despite its fundamental impor tance, Marko v’ s g roundbreaking
w ork gained recognition onl y v ery slo w l y . Hur witz beg an translating Mark o v’ s
ideas to the setting of Diophantine appro ximation. As this circle of results became
better unders tood b y more mathematicians, the translation seemed more and
more straightf or w ard. T oda y , both v ersions of Mark o v’ s theorem, the Diophantine
appro ximation v ersion and the q uadratic f or ms v ersion, are unanimously attributed
to Mark o v .
3. Mark o v’s theor em on indefinite q uadratic f orms
In this section, w e recall the quadratic f or ms v ersion of Mark o v’ s theorem.
W e consider binar y quadratic f or ms
(11) f .p ; q / D A p 2 C 2B p q C C q 2 ;
with real coefficients A , B , C . The determinant of such a f or m is the deter minant
of the cor responding symmetr ic 2  2 -matr ix,
(12) det f D AC  B 2 :

The h yperbolic g eometr y of Mark o v’ s theorem 343
Mark o v’ s theorem deals with indefinite f orms, i.e., f or ms with
det f < 0:
In this case, the quadratic pol ynomial
(13) f .x ; 1/ D Ax 2 C 2B x C C
has tw o distinct real roots,
(14)  B ˙ p  det f
A ;
pro vided A 6D 0 . If A D 0 , it mak es sense to consider  C
2B and 1 as tw o roots
in the real projectiv e line R P 1 Š R [ ¹1º . Then the f ollo wing statements are
equiv alent:
(i) The pol ynomial ( 13 ) has at least one root in Q [ ¹1º .
(ii) There e xist integ ers p and q , not both zero, such that f .p ; q / D 0 .
Con v ersel y , one ma y ask: F or whic h indefinite f or ms f does the set of values
® f .p ; q / ˇ ˇ .p ; q / 2 Z 2 ; .p ; q / 6D .0; 0/ ¯  R
sta y f ar thes t a wa y from 0 . This makes sense if w e require the f or ms f to be
nor malized to det f D  1 . Eq uiv alentl y , w e ma y ask: For whic h f or ms is the
infimum
(15) M .f / D inf
.p;q / 2 Z 2
.p;q / 6D 0
j f .p ; q / j
p j det f j
maximal? These f or ms are “most unlik e ” f or ms with at least one rational root,
f or which M .f / D 0 . K orkin and Zolotarev [ 43 ] g a v e the f ollo wing answ er:
Theorem (K orkin and Zolotare v). Let f be an indefinite binary quadr atic f or m
with r eal coefficients. If f is equiv alent to the f or m
p 2  p q  q 2 ;
then
M .f / D 2
p 5 :
Other wise,
(16) M .f /  1
p 2 :

344 B. Springb orn
Binar y quadratic f or ms f , Q
f are called equiv alent if there are integers a ,
b , c , d satisfying
j a d  b c j D 1;
such that
(17) Q
f .p; q / D f .a p C b q ; c p C d q /:
Equiv alent quadratic f or ms attain the same v alues on Z 2 .
Hur witz’ s theorem is roughly the Diophantine appro ximation v ersion of K orkin
& Zolotare v’ s theorem. The y did not publish a proof, but Mark o v obtained one
from them personall y . This w as the star ting point of his w ork on quadratic
f or ms [ 48 , 49 ], which es tablishes a bi jection betw een the classes of f or ms f or
which M .f /  2
3 and sor ted Mark o v tr iples:
Theorem (Mark o v , quadratic f or ms v ersion). (i) Le t .a ; b ; c / be any Marko v triple,
let p 1 , p 2 be int eg ers satisfying equation ( 8 ) , le t
(18) x 0 D p 2
a C b
a c  3
2 ;
let
(19) r D r 9
4  1
c 2
and let f be the indefinite q uadr atic f orm
(20) f .p ; q / D p 2  2x 0 p q C .x 2
0  r 2 / q 2 :
Then
(21) M .f / D 1
r ;
and the infimum in ( 15 ) is attained.
(ii) Conv ersely , suppose Q
f is an indefinite binary quadr atic f or m with
M . Q
f / > 2
3 :
Then ther e is a unique sorted Mar ko v triple .a ; b ; c / suc h that Q
f is equiv alent
to a multiple of t he f or m f defined by equation ( 20 ) .
N ote that the number x defined b y ( 9 ) is a root of the f or m f defined b y ( 20 ) ,
and M .f / D 2
L.x / . T able 2 lis ts representativ es f .p ; q / of the fiv e classes of
f or ms with the larg est v alues of M .f / .

The h yperbolic g eometr y of Mark o v’ s theorem 345
T able 2
The fiv e classes of indefinite quadratic f or ms whose values s ta y farthest a wa y from zero
Rank f .p ; q / M .f / a b c p 1 p 2
1 p 2  p q  q 2 2
p 5 D 0:89 : : : 1 1 1 0 1
2 p 2  2 q 2 1
p 2 D 0:70 : : : 1 1 2  1 1
3 5 p 2 C p q  11q 2 10
p 221 D 0:67 : : : 1 2 5  1 2
4 13 p 2 C 23 p q  19q 2 26
p 1517 D 0:667 : : : 1 5 13  3 2
5 29 p 2  5 p q  65q 2 58
p 7565 D 0:6668 : : : 2 5 29  7 3
R emar k 3.1. Here, too, the apparent asymmetr y betw een p 1 and p 2 is deceptiv e
(cf. R emark 2.1 (ii)). Equation ( 18 ) is eq uiv alent to
x 0 D p 1
b  a
b c C 3
2 :
4. The h yperbolic plane
W e use the half-space model of the h yperbolic plane f or all calculations. In
this section, w e summar ize some basic facts.
The h yperbolic plane is represented b y the upper half-plane of the comple x
plane,
H 2 D ¹ z 2 C j Im z > 0 º ;
where the length of a cur v e  W Œt 0 ; t 1  ! H 2 is defined as
Z t 1
t 0
j P  .t / j
Im  .t / d t :
The model is conf or mal, i.e., h yperbolic angles are equal to euclidean angles. The
group of isometr ies is the projectiv e g eneral linear g roup,
PGL 2 . R / D GL 2 . R /= R 
Š ® A 2 GL 2 . R / ˇ ˇ j det A j D 1 ¯ = ¹˙ Id º ;
where the action M W PGL 2 . R / ! Isom .H 2 / is defined as f ollo ws:
F or
A D  a b
c d  2 GL 2 . R /;

346 B. Springb orn
M A .z / D 8
ˆ
ˆ
<
ˆ
ˆ
:
a z C b
c z C d if det A>0 ,
a N z C b
c N z C d if det A<0 .
The isometr y M A preser v es or ientation if det A > 0 and re v erses or ientation
if det A < 0 . The subg roup of or ientation preser ving isometr ies is theref ore
PSL 2 . R / Š SL 2 . R /= ¹˙ Id º .
Geodesics in the h yperbolic plane are euclidean half circles or thogonal to the
real axis or euclidean v er tical lines (see Fig. 2 ). The h yperbolic distance betw een
points x C i y 0 and x C i y 1 on a v er tical geodesic is
ˇ ˇ ˇ log y 1
y 0 ˇ ˇ ˇ :
Apar t from g eodesics, horocy cles will pla y an impor tant role. The y are the
limiting case of circles as the radius tends to infinity . Equiv alentl y , horocy cles
are complete cur v es of cur v ature 1 . In the half-space model, horocy cles are
represented as euclidean circles that are tang ent to the real line, or as horizontal
lines. The center of a horocy cle is the point of tangency with the real line, or 1
f or hor izontal horocy cles.
The points on the real axis and 1 2 C P 1 are called ideal points. The y do
not belong to the h yperbolic plane, but the y cor respond to the ends of g eodesics.
All horocy cles centered at an ideal point x 2 R [ ¹1º intersect all g eodesics
ending in x or thogonally . In the proof of Proposition 8.1 , w e will use the f act
that tw o horocy cles centered at the same ideal point are eq uidistant curv es.
x C i y 0
x C i y 1
log y 0
y 1
g eodesics
horocy cles
p 0 2
1
q 2
p
q
h.p 0 ; 0/
h.p ; q /
Figure 2
Geodesics and horocy cles

The h yperbolic g eometr y of Mark o v’ s theorem 347
5. Dictionary : Horocy cle – 2D v ector
W e assign a horocy cle h.p; q / to e v er y .p ; q / 2 R 2 n ¹ .0; 0/ º as f ollo w s (see
Fig. 2 ):
 F or q 6D 0 , let h.p; q / be the horocy cle at p
q with euclidean diameter 1
q 2 .
 Let h.p ; 0/ be the horocy cle at 1 at height p 2 .
The map .p ; q / 7! h.p; q / from R 2 n ¹ 0 º to the space of horocy cles is sur jectiv e
and tw o-to-one, mapping ˙ .p ; q / to the same horocy cle. The map is equiv ariant
with respect to the PGL 2 . R / -action [ 24 , p. 665]. More precisel y :
Proposition 5.1 (Eq uiv ar iance). F or A 2 GL 2 . R / satisfying j det A j D 1 and f or
v 2 R 2 n ¹ 0 º , the hyperbolic isome tr y M A maps the hor ocy cle h.v / to h.Av / .
Pr oof. This can of course be sho wn b y direct calculation. T o simplify the
calculations, note that e v er y isometr y of H 2 can be represented as a composition
of isometr ies of the f ollo wing types:
(22) z 7! z C b ; z 7! z ; z 7!  N z ; z 7! 1
N z
(where b 2 R ,  2 R >0 ). The cor responding nor malized matr ices are
(23) 1 b
0 1 ! ;  1
2 0
0   1
2 ! ;  1 0
0 1 ! ; 0 1
1 0 ! :
(The first tw o maps preser v e or ientation, the other tw o re v erse it.) It is theref ore
enough to do the simpler calculations f or these maps. (For the in v ersion, Fig. 3
indicates an alter nativ e g eometr ic ar gument, just f or fun.)
0 q
p 1 p
q
1
2p 2
1
2 q 2
Figure 3
Horocy cle h.p ; q / and image under in v ersion z 7! 1
N z

348 B. Springb orn
6. Signed dist ance of tw o horocy cles
The signed distance d .h 1 ; h 2 / of horocy cles h 1 , h 2 is defined as f ollo ws (see
Fig. 4 ):
 If h 1 and h 2 are centered at different points and do not intersect,
then d .h 1 ; h 2 / is the length of the g eodesic segment connecting the horocy cles
and or thogonal to both. (This is just the h yperbolic distance betw een the
horocy cles.)
 If h 1 and h 2 do intersect, then d .h 1 ; h 2 / is the length of that geodesic
segment, tak en neg ativ e. (If h 1 and h 2 are tang ent, then d .h 1 ; h 2 / D 0 .)
 If h 1 and h 2 ha v e the same center , then d .h 1 ; h 2 / D 1 .
d > 0
d < 0
Figure 4
The signed distance of horocy cles
R emar k 6.1. If horocy cles h 1 , h 2 ha v e the same center , the y are equidis tant
cur v es with a w ell defined finite dis tance. But their signed dis tance is defined to
be 1 . Other wise, the map .h 1 ; h 2 / 7! d .h 1 ; h 2 / w ould not be continuous on
the diagonal.
Proposition 6.2 (Signed dis tance of horocy cles). The signed distance of tw o
hor ocy cles h 1 D h.p 1 ; q 1 / and h 2 D h.p 2 ; q 2 / is
(24) d .h 1 ; h 2 / D 2 log j p 1 q 2  p 2 q 1 j :
Pr oof. It is easy to deriv e equation ( 24 ) if one horocy cle is centered at 1 (see
Fig. 2 ). T o pro v e the g eneral case, apply the h yperbolic isometr y
M A .z / D 1
z  p 1
q 1
; A D 0 1
1  p 1
q 1 !
that maps one horocy cle center to 1 and use Proposition 5.1 .

The h yperbolic g eometr y of Mark o v’ s theorem 349
7. F ord cir cles and F are y tessellation
Figure 5 sho ws the horocy cles h.p ; q / with integ er parameters .p; q / 2 Z 2 .
There is an infinite f amil y of such integ er horocy cles centered at each rational
number and at 1 . (Only the lo w est horocy cle centered at 1 is sho wn to sa v e
space.) Integ er horocy cles h.p 1 ; q 1 / and h.p 2 ; q 2 / with different centers p 1
q 1 6D p 2
q 2
do not intersect. This f ollo ws from Proposition 6.2 , because p 1 q 2  p 2 q 1 is a
non-zero integ er . The y touc h if and onl y if p 1 q 2  p 2 q 1 D ˙ 1 . This can happen
onl y if both .p 1 ; q 1 / and .p 2 ; q 2 / are copr ime, that is, if p 1
q 1 and p 2
q 2 are reduced
fractions representing the respectiv e horocy cle centers.
Figure 6 sho ws the horocy cles h.p ; q / with integ er and copr ime parame-
ters .p ; q / . The y are called F or d cir cles . There is e xactl y one Ford circle centered
at each rational number and at 1 . If one connects the ideal centers of tang ent
F ord circles with g eodesics, one obtains the F ar ey tessellation , whic h is also
sho wn in the figure. The F are y tessellation is an ideal tr iangulation of the h y -
perbolic plane with v er te x set Q [ ¹1º . (A thorough treatment can be f ound
in [ 7 ].)
 1 0 1 2
1
2
1
3
2
3
1
4
3
4
1
5
2
5
3
5
4
5
Figure 5
Horocy cles h.p ; q / with integer parameters .p ; q / 2 Z 2
 1 0 1 2
1
2
1
3
2
3
1
4
3
4
1
5
2
5
3
5
4
5
Figure 6
F ord circles and F arey tessellation

350 B. Springb orn
W e will see that Mark o v tr iples cor respond to ideal tr iangulations of the h y -
perbolic plane (as univ ersal co v er of the modular tor us), and .1; 1; 1/ cor responds
to the F are y tessellation (Sec. 11 ). The F are y tessellation also comes up when one
considers the minima of definite q uadratic f or ms (Sec. 16 ).
8. Signed dist ance of a horocy cle and a g eodesic
F or a horocy cle h and a g eodesic g , the signed distance d .h; g / is defined
as f ollo ws (see F ig. 7 ):
 If h and g do not intersect, then d .h; g / is the length of the g eodesic
segment connecting h and g and orthogonal to both. (This is just the
h yperbolic distance betw een h and g .)
 If h and g do intersect, then d .h; g / is the length of that g eodesic segment,
tak en neg ativ e.
 If h and g are tang ent then d .h; g / D 0 .
 If g ends in the center of h then d .h; g / D 1 .
h
g
h
g
d > 0
d < 0
x 1 x 2 x 1 x 2
Figure 7
The signed distance d D d .h; g / of a horocy cle h and a g eodesic g
An equation f or the signed distance to a v er tical geodesic is particularl y easy
to der iv e:
Proposition 8.1 (Signed dis tance to a v er tical g eodesic). Consider a hor ocycle
h D h.p ; q / wit h q 6D 0 and a v er tical g eodesic g fr om x 2 R t o 1 . Their
signed distance is
(25) d .h; g / D log  2 q 2 ˇ ˇ ˇ x  p
q ˇ ˇ ˇ  :
Pr oof. See F ig. 8 .

The h yperbolic g eometr y of Mark o v’ s theorem 351
x p
q
d
d
1
q 2
2 ˇ ˇ x  p
q ˇ ˇ
g
h
Figure 8
Signed distance of horocy cle h D h.p ; q / and v er tical geodesic g
Equation ( 25 ) sugg ests a g eometr ic inter pretation of Hurwitz’ s theorem and the
Diophantine appro ximation v ersion of Mark o v’ s theorem: A fraction p
q satisfies
inequality ( 2 ) if and onl y if
(26) d  h.p ; q /; g  <  log 
2 :
The f ollo wing section contains a proof of Hur witz’ s theorem based on this
obser v ation. An eq uation f or the signed distance to a g eneral g eodesic will be
presented in Proposition 10.1 .
9. Proof of Hurwitz’ s theorem
Let x be an ir rational number and let g be the v er tical g eodesic from x
to 1 . By Proposition 8.1 , par t (i) of Hur witz’ s theorem is equiv alent to the
statement:
Infinitel y man y Ford circles h satisfy
(27) d .h; g / <  log p 5
2 :
This f ollo ws from the f ollo wing lemma. Let us sa y that the midpoint of an
edg e of the F are y tessellation is the point where the horocy cles centered at its
ends meet (see Fig. 6 ). A ccordingl y , w e sa y that a geodesic bisects an edg e of
the F are y tessellation if it passes through the midpoint of the edg e (see F ig. 9 ).
Lemma 9.1. Suppose a g eodesic g cr osses an ideal triang le T of the F ar ey
tessellation. If g is one of the thr ee g eodesics bisecting tw o sides of T , t hen
d .h; g / D  log p 5
2

352 B. Springb orn
1
2  p 5
2 0 1
2 1 1
2 C p 5
2 D ˆ
1
p 5
2
d
g 1
g
p 5
2
Figure 9
Geodesic g 1 bisecting the tw o v er tical sides of the
tr iangle 0; 1; 1 , and g eodesic g from ˆ to 1
f or all thr ee F or d cir cles h at the v ertices of T . Other wise, inequality ( 27 ) holds
f or at least one of these thr ee F or d circles.
Pr oof of Lemma 9.1 . This is the simples t case of Propositions 13.2 and 13.4 , and
easy to pro v e independentl y . N ote that it is enough to consider the ideal tr iangle
0 , 1 , 1 , and g eodesics intersecting its tw o v er tical sides (see Fig. 9 ).
T o deduce par t (i) of Hurwitz’ s theorem, note that since x is ir rational,
the g eodesic g from x to 1 passes through infinitel y man y tr iangles of the
F are y tessellation. F or eac h of these triangles, at least one of its F ord circles
h satisfies ( 27 ) , b y Lemma 9.1 . (The g eodesic g does not bisect tw o sides of
an y F are y tr iangle. Other wise, g w ould bisect tw o sides of all F are y tr iangles
it enters; see Fig. 9 , where the ne xt tr iangle is sho wn with dashed lines. This
contradicts g ending in the v er te x 1 of the F are y tessellation.)
F or consecutiv e tr iangles that g crosses, the same horocy cle ma y satisfy ( 27 ) .
But this can happen onl y finitel y man y times (other wise x w ould be rational),
and then the g eodesic will ne v er ag ain intersect a triangle incident with this
horocy cle. Hence, infinitely man y F ord circles satisfy ( 27 ) , and this completes
the proof of par t (i).
T o pro v e par t (ii) of Hur witz’ s theorem, w e ha v e to sho w that f or
x D ˆ and  > 0;
onl y finitel y man y F ord circles h satisfy
(28) d .h; g / <  log p 5
2   ;
where g is the g eodesic from ˆ to 1 .

The h yperbolic g eometr y of Mark o v’ s theorem 353
T o this end, let g 1 be the g eodesic from ˆ D 1
2 .1 C p 5/ to 1
2 .1  p 5/ , see
Fig. 9 . F or ev er y Ford circle h ,
d .h; g 1 /   log p 5
2 :
Indeed, the distance is eq ual to  log p 5
2 f or all Ford circles that g 1 intersects,
and positiv e f or all others.
Because the g eodesics g and g 1 con v er g e at the common end ˆ , there is
a point P 2 g such that all F ord circles h intersecting the ra y from P to ˆ
satisfy
j d .g ; ˆ/  d .g 1 ; ˆ/ j <  ;
and hence
d .g ; ˆ/   log p 5
2   :
On the other hand, the complementar y ra y of g , from P to 1 , intersects onl y
finitel y man y Ford circles. Hence, onl y finitel y man y Ford circles satisfy ( 28 ) ,
and this completes the proof of par t (ii).
R emar k 9.2. The gis t of the abo v e proof is deducing Hur witz’ s theorem from
the f act that the g eodesic g from an ir rational number x to 1 crosses infinitel y
man y F are y tr iangles. A w eak er s tatement f ollo ws from the observation that g
crosses infinitel y man y edg es. Since eac h edg e has tw o touching F ord circles at
the ends, a crossing g eodesic intersects at leas t one of them. Hence there are
infinitel y man y fractions satisfying ( 2 ) with  D 2 . In f act, at leas t one of an y
tw o consecutiv e continued fraction appro ximants satisfies this bound. This result
is due to V ahlen [ 58 , p. 41] [ 72 ]. The con v erse is due to Leg endre [ 45 ] and 65
y ears older: If a fraction satisfies ( 2 ) with  D 2 , then it is a continued fraction
appro ximant. A geometric proof using Ford circles is mentioned b y Speiser [ 70 ]
(see Sec. 1 ).
10. Dictionary : Geodesic – indefinite f orm
W e assign a geodesic g .f / to ev er y indefinite binar y quadratic f or m f
with real coefficients as f ollo ws: T o the f or m f with real coefficients A , B ,
C as in ( 11 ) , w e assign the geodesic g .f / that connects the zeros of the
pol ynomial ( 13 ) . (If A D 0 , one of the zeros is 1 , and g .f / is a v er tical
g eodesic.) The map f 7! g .f / from the space of indefinite f or ms to the space
of g eodesics is
 surjectiv e and man y-to-one: g .f / D g . Q
f / , Q
f D  f f or some  2 R  .

354 B. Springb orn
 equiv ar iant with respect to the left GL 2 . R / -actions:
f f ı A  1
g .f / M A g .f / D g .f ı A  1 /
A
g A 2 GL 2 . R / g
M A
Proposition 10.1. The signed dis tance of the hor ocycle h.p ; q / and the g eo-
desic g .f / is
(29) d  h.p ; q /; g .f /  D log j f .p ; q / j
p  det f :
Pr oof. F irs t, consider the case of hor izontal horocy cles ( q D 0 ). If g .f / is
a v er tical geodesic ( f .p ; 0/ D 0 ), equation ( 29 ) is immediate. Otherwise, note
that p 2 p  det f = j f .p ; 0/ j is half the distance betw een the zeros ( 14 ) , hence the
height of the g eodesic.
The g eneral case reduces to this one: F or an y A 2 GL 2 . R / with j det A j D 1
and A  p
q  D  Q p
0  ,
d  h.p ; q /; g .f /  D d  M A h.p ; q /; M A g .f /  D d  h. Q p ; 0/; g .f ı A  1 / 
D log j .f ı A  1 /. Q p ; 0/ j
p  det .f ı A  1 / D log j f .p ; q / j
p  det f :
Equation ( 29 ) sugg ests a g eometr ic inter pretation of the quadratic f or ms v ersion
of Mark o v’ s theorem, and it is easy to pro v e most of K orkin & Zolotare v’ s
theorem (just replace ineq uality ( 16 ) with M .f / < 2
p 5 ) b y adapting the proof
of Hur witz’ s theorem in Sec. 9 . T o obtain the complete Mark o v theorem, more
h yperbolic geometry is needed. This this is the subject of the f ollo wing sections.
11. Decorated ideal triangles
In this and the f ollo wing section, w e re vie w some basic f acts from P enner ’ s
theor y of decorated T eic hmüller spaces [ 55 , 56 ]. The material of this section, up
to and including equation ( 30 ) is enough to treat crossing g eodesics in Sec. 13 .
Ptolem y’ s relation is needed f or the g eometr ic inter pretation of Mark o v’ s equation
in Sec. 12 .
An ideal triang le is a closed region in the h yperbolic plane that is bounded
b y three geodesics (the sides ) connecting three ideal points (the v ertices ). Ideal
tr iangles ha v e dihedral symmetry , and an y tw o ideal tr iangles are isometr ic. That
is, f or an y pair of ideal triangles and an y bi jection betw een their v er tices, there is

The h yperbolic g eometr y of Mark o v’ s theorem 355
˛ 3
˛ 1
˛ 2
c 3
c 1 c 2
h 1
h 2
h 3
˛ 3
˛ 1
˛ 2
c 3
c 1 c 2
0 1
i 1 C i
Figure 10
Decorated ideal tr iangle in the P oincaré disk
model (left) and in the half-plane model (r ight)
a unique h yperbolic isometr y that maps one to the other and respects the v er te x
matching. A decor ated ideal triang le is an ideal tr iangle tog ether with a horocy cle
at each v er te x (F ig. 10 ).
Consider a g eodesic decorated with tw o horocy cles h 1 , h 2 at its ends (f or
e xample, a side of an ideal tr iangle). Let the truncated lengt h of the decorated
g eodesic be defined as the signed dis tance of the horocy cles (Sec. 6 ),
˛ D d .h 1 ; h 2 /;
and let its w eight be defined as
a D e ˛=2 :
(W e will often use Greek letters f or tr uncated lengths and Latin letters f or w eights.
The w eights are usually called  -lengths .)
An y tr iple .˛ 1 ; ˛ 2 ; ˛ 3 / 2 R 3 of tr uncated lengths, or , equiv alently , an y tr iple
.a 1 ; a 2 ; a 3 / 2 R 3
>0 of w eights, deter mines a unique decorated ideal tr iangle up to
isometr y .
Consider a decorated ideal tr iangle with tr uncated lengths ˛ k and w eights a k .
Its horocy cles intersect the tr iangle in three finite arcs. Denote their h yperbolic
lengths b y c k (see Fig. 10 ). The truncated side lengths deter mine the horocy clic
arc lengths, and vice v ersa, via the relation
(30) c k D a k
a i a j D e 1
2 .  ˛ i  ˛ j C ˛ k / ;
where .i ; j ; k / is a permutation of .1; 2 ; 3/ . (F or a proof, contemplate F ig. 10 .)

356 B. Springb orn
N o w consider a decorated ideal quadr ilateral as sho wn in Fig. 11 . It can be
decomposed into tw o decorated ideal tr iangles in tw o wa y s. The six w eights a ,
b , c , d , e , f are related b y the Ptolemy relation
(31) e f D a c C b d :
It is straightf or w ard to der iv e this equation using the relations ( 30 ) .
a
b
c
d e
f
Figure 11
Ptolem y relation
a
a 0
b
b
c
c
Figure 12
T r iangulations T and T 0 of a punctured torus
12. T riangulations of the modular torus and Mark o v’s eq uation
In this section, w e revie w P enner ’ s [ 55 , 56 ] geometric inter pretation of
Mark o v’ s equation ( 4 ) , which is summarized in Prop. 12.1 . The in v olutions  k
w ere defined in Sec. 2 , see equation ( 7 ) . The modular torus is the orbit space
M D H 2 = G;
where G is the group of or ientation preser ving h yperbolic isometries generated
b y
(32) A.z / D z  1
 z C 2 ; B .z / D z C 1
z C 2 :
Figure 13 sho ws a fundamental domain. The g roup G is the commutator subgroup
of the modular group PSL 2 . Z / , and the only subgroup of PSL 2 . Z / that has a once
punctured tor us as orbit space. It is a nor mal subgroup of PSL 2 . Z / with inde x
six, and the quotient group PSL 2 . Z /= G is the g roup of or ientation preser ving
isometr ies of the modular tor us M . It is also symmetric with respect to six
reflections, so the isometr y group has in total tw el v e elements.

The h yperbolic g eometr y of Mark o v’ s theorem 357
 1 0 1
B A
Figure 13
The modular tor us
Proposition 12.1 (Mark o v tr iples and ideal tr iangulations). (i) A triple  D .a ; b ; c /
of positiv e integ ers is a Marko v triple if and only if ther e is an ideal triangulation
of the decor ated modular torus whose thr ee edg es hav e the w eights a , b , and
c . This triangulation is unique up t o the 12 -f old symme tr y of the modular torus.
(ii) If T is an ideal triangulation of the decor ated modular torus with edg e
w eights  D .a ; b ; c / , and if T 0 is an ideal triangulation obtained fr om T by
per f orming a sing le edg e flip, then the edg e w eights of T 0 ar e  0 D  k  , wit h
k 2 ¹ 1; 2 ; 3 º depending on whic h edg e w as flipped.
T o understand the logical connections, it mak es sense to consider not onl y the
modular tor us but arbitrar y once punctured h yperbolic tor i.
A once punctur ed hyperbolic torus is a tor us with one point remo v ed, equipped
with a complete metr ic of constant curv ature  1 and finite v olume. F or e xample,
one obtains a once punctured h yperbolic tor us by gluing tw o cong r uent decorated
ideal tr iangles along their edg es in suc h a w a y that the horocy cles fit tog ether .
Con v ersel y , e v er y ideal tr iangulation of a h yperbolic torus with one puncture
decomposes it into tw o ideal tr iangles.
A decor at ed once punctur ed hyperbolic torus is a once punctured h yperbolic
tor us tog ether with a c hoice of horocy cle at the cusp. Thus, a tr iple of w eights
.a ; b ; c / 2 R 3
>0 deter mines a decorated once punctured h yperbolic torus up to
isometr y , together with an ideal triangulation. Con v ersel y , a decorated once
punctured h yperbolic tor us together with an ideal triangulation deter mines such
a tr iple of edg e w eights.
Consider a decorated once punctured h yperbolic tor us with an ideal tr iangu-
lation T with edg e w eights .a ; b ; c / 2 R 3
>0 . By equation ( 30 ) , the total length of
the horocy cle is

358 B. Springb orn
` D 2  a
b c C b
c a C c
a b  :
This equation is eq uiv alent to
a 2 C b 2 C c 2 D `
2 a b c :
Thus, the w eights satisfy Marko v’ s equation ( 4 ) (not considered as a Diophantine
equation) if and onl y if the horocy cle has length ` D 6 . From no w on, w e assume
that this is the case: W e decorate all once punctured h yperbolic tor i with the
horocy cle of length 6 .
Let T 0 be the ideal tr iangulation obtained from T b y flipping the edge
with w eight a , i.e., b y replacing this edge with the other diagonal in the ideal
quadrilateral f or med b y the other edg es (see F ig. 12 ). By equation ( 6 ) and Ptolem y’ s
relation ( 31 ) , the edg e w eights of T 0 are .a 0 ; b ; c / D  1 .a ; b ; c / . Of course, one
obtains analogous equations if a different edg e is flipped.
The modular tor us M , decorated with a horocy cle of length 6 , is obtained b y
gluing tw o decorated ideal tr iangles with w eights .1; 1; 1/ . Lifting this tr iangulation
and decoration to the h yperbolic plane, one obtains the F are y tessellation with
F ord circles (F ig. 6 ). This implies that f or e v er y Mark o v tr iple .a ; b ; c / there is
an ideal tr iangulation of the decorated modular tor us with edg e w eights a , b , c .
T o see this, f ollo w the path in the Mark o v tree leading from .1; 1; 1/ to .a; b ; c /
and per f or m the cor responding edg e flips on the projected F are y tessellation.
On the other hand, the flip graph of a complete hyperbolic surf ace with
punctures is also connected [ 34 ] [ 54 , p. 36ff]. The flip g r aph has the ideal
tr iangulations as v er tices, and edges connect triangulations related b y a single
edg e flip. (Since w e are onl y interested in a once punctured torus, in v oking this
g eneral theorem is some what of an o v erkill.) This implies the con v erse s tatement:
If a , b , c are the w eights of an ideal triangulation of the modular tor us, then
.a ; b ; c / is a Mark o v triple.
N ote that there is onl y one ideal triangulation of the modular tor us with
w eights .1; 1; 1/ , i.e., the triangulation that lifts to the F are y tessellation. The
symmetr ies of the modular tor us per mute its edg es. Since the Mark o v tree and
the flip graph are isomor phic, this implies that tw o tr iangulations with the same
w eights are related by an isometry of the modular tor us. Altog ether , one obtains
Proposition 12.1 .
13. Geodesics crossing a decorated ideal triangle
F or the proof of Mark o v’ s theorem in Sec. 15 , we need to kno w ho w f ar a
g eodesic crossing a decorated ideal triangle can sta y a w a y from the horocy cles at

The h yperbolic g eometr y of Mark o v’ s theorem 359
the v er tices. T o pro v e Hur witz’ s theorem (see Sec. 9 ), it was enough to consider
a tr iangle decorated with pair wise tang ent horocy cles. In this section, w e consider
the g eneral case, more precisel y , the f ollo wing g eometr ic optimization problem:
Problem 13.1. Giv en a decorated ideal tr iangle with tw o sides, sa y a 1 and a 2 ,
designated as “legs ”, and the third side, sa y a 3 , designated as “base ”. F ind , among
all g eodesics intersecting both legs, a g eodesic that maximizes the minimum of
signed distances to the three horocy cles at the v er tices.
It mak es sense to consider the cor responding optimization problem f or
euclidean tr iangles: Whic h straight line crossing tw o giv en legs has the larg est
distance to the v er tices? The answer depends on whether or not an angle at the
base is obtuse. F or decorated ideal tr iangles, the situation is completel y analogous.
W e sa y that a g eodesic bisects a side of a decorated ideal triangle if it intersects
the side in the point at equal dis tance to the tw o horocy cles at the ends of the
side.
Proposition 13.2. Consider a decor at ed ideal triang le with hor ocy cles h 1 , h 2 ,
h 3 , and let a 1 , a 2 , a 3 deno t e bo th t he sides and their w eights (see F ig. 14 f or
notation).
(i) If
(33) a 2
1  a 2
2 C a 2
3 and a 2
2  a 2
1 C a 2
3 ;
then t he g eodesic g bisecting the sides a 1 and a 2 is the unique solution of
Pr oblem 13.1 .
(ii) If, f or .j ; k / 2 ¹ .1; 2/; .2 ; 1/ º ,
(34) a 2
j  a 2
k C a 2
3 ;
then t he perpendicular bisector g 0 of side a k is the unique solution of
Pr oblem 13.1 . In t his case, the minimal dis tance is attained f or h j and h 3 ,
(35) d .h j ; g 0 / D d .h 3 ; g 0 / D ˛ k
2  d .h k ; g 0 /:
In the proof of Mark o v’ s theorem (Sec. 15 ), the base a 3 will alw a y s be a
larg est side, so onl y par t (i) of Proposition 13.2 is needed. W e will also need some
equations f or the geodesic bisecting tw o sides, which w e collect in Proposition 13.4 .
Pr oof of Pr oposition 13.2 . 1. The geodesic g has equal distance from all three
horocy cles. Indeed, because of the 180 ı rotational symmetr y around the inter -
section point, an y geodesic bisecting a side has eq ual distance from the tw o
horocy cles at the ends.

360 B. Springb orn
v 2
v 1
v 3 D 1
a 1
a 2
a 3
h 2
h 1
h 3
g P 2
P 1
P 3
c 2
c 1
c 3
s 2
s 2
s 1
s 1
s 3
s 3
x 1 x 0 x 2
r
1
1
a 1
1
a 2
1
v 2
v 1
v 3 D 1
h 2
h 1
h 3
g
P 2
P 1
P 3
a 1
a 2
a 3
c 3
s 2
s 1
c 1
s 3
s 2
c 2 s 1
s 3
Figure 14
Decorated ideal tr iangle (shaded) and g eodesic g through the midpoints of sides a 1
and a 2 . Lef t : Inequalities ( 33 ) are strictly satisfied and P 3 lies strictly betw een
P 1 and P 2 . (The height marks on the r ight mar gin belong to the proof of
Proposition 13.4 .) Right : a 2
1 > a 2
2 C a 2
3 and P 1 lies strictl y betw een P 3 and P 2 .

The h yperbolic g eometr y of Mark o v’ s theorem 361
2. F or k 2 ¹ 1; 2 ; 3 º let P k be the f oot of the per pendicular from v er te x v k to
the g eodesic g bisecting a 1 and a 2 (see Fig. 14 ). If P 3 lies strictly betw een P 1
and P 2 (as in Fig. 14 , left), then g is the unique solution of Problem 13.1 . An y
other g eodesic crossing a 1 and a 2 also crosses at least one of the ra ys from P k
to v k , and is theref ore closer to at least one of the horocy cles.
3. If P 1 lies s tr ictl y betw een P 3 and P 2 (as in F ig. 14 , r ight) then the uniq ue
solution of Problem 13.1 is the per pendicular bisector of a 2 . Its signed distance
to the horocy cles h 1 and h 3 is half the tr uncated length of side a 2 . An y other
g eodesic crossing a 2 is closer to at least one of its horocy cles. The signed distance
of g and the horocy cle h 2 is lar ger . The case when P 1 lies strictly betw een P 3
and P 2 is treated in the same w a y .
5. If P 2 D P 3 (or P 1 D P 3 ) then the geodesic g with equal dis tance to all
horocy cles is simultaneously the perpendicular bisector of side a 2 (or a 1 ).
6. It remains to sho w that the order of the points P k on g depends on whether
the w eights satisfy the inequalities ( 33 ) or one of the inequalities ( 34 ) . T o this
end, let s 1 be the distance from the side a 1 to the ra y P 3 v 3 , measured along the
horocy cle h 3 in the direction from a 1 to a 2 . Similar l y , let s 2 be the distance
from the side a 2 to the ra y P 3 v 3 , measured along the horocy cle h 3 in the
direction from a 2 to a 1 . So s 1 and s 2 are both positiv e if and onl y if P 3 lies
strictl y betw een P 1 and P 2 . But if, f or e xample, P 1 lies betw een P 3 and P 2 as
in Fig. 14 , right, then s 2 < 0 . By symmetr y , s 1 is also the distance from a 1 to
P 2 v 2 , measured along h 2 in the direction a w a y from a 3 . Similar l y , s 2 is also
the distance betw een a 2 and P 1 v 1 along h 1 . Finall y , let s 3 > 0 be the equal
distances betw een a 3 and P 1 v 1 along h 1 , and betw een a 3 and P 2 v 2 along h 2 .
N o w
c 1 D  s 2 C s 3 ; c 2 D  s 1 C s 3 ; c 3 D s 1 C s 2
implies
(36) 2 s 1 D c 1  c 2 C c 3
( 30 )
D a 1
a 2 a 3  a 2
a 3 a 1 C a 3
a 1 a 2 D a 2
1  a 2
2 C a 2
3
a 1 a 2 a 3
and similar l y
2 s 2 D  a 2
1 C a 2
2 C a 2
3
a 1 a 2 a 3
:
Hence, P 3 lies in the closed inter v al betw een P 1 and P 2 if and onl y if
inequalities ( 33 ) are satisfied. The other cases are treated similar ly .
R emar k 13.3. The abo v e proof of Proposition 13.2 is nicel y intuitiv e. A more
anal ytic proof ma y be obtained as f ollo w s. Firs t, sho w that f or all geodesics

362 B. Springb orn
intersecting a 1 and a 2 , the signed distances u 1 , u 2 , u 3 to the horocy cles satisfy
the equation
(37) .c 1 u 1 C c 2 u 2 C c 3 u 3 / 2  4 c 1 c 2 u 1 u 2  4 D 0
It mak es sense to consider the special case a 1 D a 2 D a 3 D 1 firs t, because
the g eneral eq uation ( 37 ) can easil y be deriv ed from the simpler one. Then
consider the necessar y conditions f or a local maximum of min .u 1 ; u 2 ; u 3 / under
the constraint ( 37 ) : If a maximum is attained with u 1 D u 2 D u 3 , then the three
par tial der iv ativ es of the left hand side of ( 37 ) are all  0 or all  0 . If a
maximum is attained with u 1 D u 2 < u 3 , then this sign condition holds f or the
first tw o der ivativ es, and similarl y f or the other cases.
Proposition 13.4. Le t g be t he g eodesic bisecting sides a 1 and a 2 of a decorat ed
ideal triang le as sho wn in F ig. 14 . (Inequalities ( 33 ) may hold or no t.) Then the
common signed distance of g and the hor ocy cles is
d .h 1 ; g / D d .h 2 ; g / D d .h 3 ; g / D  log r ;
wher e
(38) r D s ı 2
4  1
a 2
3
;
and ı is the sum of t he lengths of t he hor ocyclic ar cs,
(39) ı D c 1 C c 2 C c 3 D a 1
a 2 a 3 C a 2
a 3 a 1 C a 3
a 1 a 2
:
Mor eov er , suppose the v ertices are
(40) v 1 < v 2 ; v 3 D 1 ;
and the hor ocycle h 3 has height 1 . Then the ends x 1;2 of g ar e
(41) x 1;2 D x 0 ˙ r ;
wher e
(42) x 0 D v 2 C a 2
a 3 a 1  ı
2
Pr oof. Assuming ( 40 ) and h 3 D h.1; 0/ , let x 0 D v 2  s 1 . Then the proposition
f ollo ws from ( 36 ) , some easy h yperbolic g eometr y , Pythagoras ’ theorem, and
simple alg ebra (see F ig. 14 ).

The h yperbolic g eometr y of Mark o v’ s theorem 363
14. Simple closed g eodesics and ideal ar cs
In this section, w e collect some topological facts about simple closed g eodesics
and ideal arcs that w e will use in the proof of Mark o v’ s theorem (Sec. 15 ). The y
are probabl y w ell kno wn, but w e indicate proofs f or the reader’ s con v enience.
An ideal ar c in a complete h yperbolic sur f ace with cusps is a simple g eodesic
connecting tw o punctures or a puncture with itself. The edg es of an ideal
tr iangulation are ideal arcs, and e v er y ideal arc occurs in an ideal tr iangulation.
(In f act, ideal triangulations are e xactl y the maximal sets of non-intersecting ideal
arcs.) Here, w e are only interes ted in a once punctured h yperbolic tor us. In this
case, e v er y ideal tr iangulation containing a fix ed ideal arc can be obtained from
an y other such tr iangulation b y repeatedly flipping the remaining tw o edges. Ideal
arcs pla y an impor tant role in the f ollo wing section because the y are in one-
to-one cor respondence with the simple closed g eodesics (Proposition 14.1 ), and
the simple closed g eodesics are the g eodesics that sta y farthest a wa y from the
puncture (Proposition 15.1 ).
Proposition 14.1. Consider a fixed once punctur ed hyperbolic t orus.
(i) F or ev er y ideal ar c c , ther e is a unique simple closed g eodesic g that does
not int ersect c .
(ii) Ev er y o ther g eodesic not int er secting c has either tw o ends in the punctur e,
or one end in the punctur e and the o ther end appr oac hing t he closed g eodesic
g .
(iii) If a , b , c ar e the edg es of an ideal triangulation T , then t he simple closed
g eodesic g that does no t int er sect c int er sects eac h of the tw o triang les of
T in a g eodesic segment bisecting the edg es a and b .
(iv) F or ev er y simple closed g eodesic g , ther e is a unique ideal ar c c that does
not int ersect g .
R emar k 14.2. Speaking of edg e midpoints implies an (arbitrar y) c hoice of a
horocy cle at the cusp. In fact, the edg e midpoints of a tr iangulated once punctured
tor us are dis tinguished without an y c hoice of triangulation. The y are the three
fix ed points of an or ientation preserving isometr ic in v olution. Ev ery ideal arc
passes through one of these points.
Pr oof. (i) Cut the torus along the ideal arc c . The result is a h yperbolic cy linder
as sho wn in Fig. 15 (left). Both boundar y curv es are complete g eodesics with
both ends in the cusp, which is no w split in tw o. There is up to or ientation
a unique non-trivial free homotop y class that contains simple cur v es, and
this class contains a unique simple closed g eodesic.

364 B. Springb orn
c
g
c
g
c
g
Figure 15
Cutting a punctured tor us along an ideal arc
(left) and along a simple closed g eodesic (right).
(ii) Consider the univ ersal co v er of the cy linder in the h yperbolic plane.
(iii) An ideal triangulation of a once punctured tor us is symmetr ic with respect to
a 180 ı rotation around the edg e midpoints. (This is the in v olution mentioned
in R emark 14.2 .) It sw aps the g eodesic segments bisecting edges a and b in
the tw o ideal tr iangles, so the y connect smoothl y . Hence the y f orm a simple
closed g eodesic, whic h does not intersect c .
(iv) Cut the torus along the simple closed geodesic g . The result is a cy linder
with a cusp and tw o geodesic boundary circles, as sho wn in F ig. 15 (right).
Fill the puncture and tak e it as base point f or the homotop y group. There
is up to or ientation a unique non-trivial homotop y class containing simple
closed cur v es and this class contains a unique ideal arc.
15. Proof of Mar k o v’s theor em
In this section, w e put the pieces together to pro v e both v ersions of
Mark o v’ s theorem. The quadratic f or ms v ersion f ollo w s from Proposition 15.1 .
The Diophantine appro ximation v ersion f ollo w s from Proposition 15.1 together
with Proposition 15.2 .
T w o geodesics in the h yperbolic plane are GL 2 . Z / -r elat ed if, f or some
A 2 GL 2 . Z / , the h yperbolic isometry M A maps one to the other .
Proposition 15.1. Let g be a comple te g eodesic in the hyperbolic plane, and le t
 .g / be its pr ojection to the modular t orus. Then the f ollowing t hr ee statements
ar e equiv alent :

The h yperbolic g eometr y of Mark o v’ s theorem 365
(a)  .g / is a simple closed g eodesic.
(b) Ther e is a Marko v triple .a; b ; c / so that f or one (hence any) choice of
integ ers p 1 , p 2 satisfying ( 8 ) , t he g eodesic g is GL 2 . Z / -relat ed to the
g eodesic ending in x 0 ˙ r wit h x 0 and r defined by ( 18 ) and ( 19 ) .
(c) The g r eatest lo w er bound f or t he signed distances of g and a F or d cir cle is
g r eater than  log 3
2 .
If g satisfies one (hence all) of the s tat ements (a) , (b) , (c) , then
(d) the minimal signed dis tance of g and a F or d cir cle is  log r ,
(e) among all Marko v triples .a ; b ; c / that v erify (b) , ther e is a uniq ue sorted
Marko v triple.
Pr oof. “ (a) ) (b) ”: If  .g / is a simple closed g eodesic, then there is a unique
ideal arc c not intersecting  .g / (Proposition 14.1 (iv)). Pick an ideal triangulation
T of the modular tor us that contains c , and let a and b be the other edg es.
By Proposition 12.1 , .a ; b ; c / is a Mark o v tr iple. (W e use the same letters to
denote both ideal arcs and their w eights.) The g eodesic  .g / intersects each
of the tw o tr iangles of T in a geodesic segment bisecting the edg es a and b
(Proposition 14.1 (iii)).
N o w let p 1 , p 2 be integ ers satisfying ( 8 ) and consider the decorated ideal
tr iangle in H 2 with v er tices
(43) v 1 D p 1
b ; v 2 D p 2
a ; v 3 D 1 ;
and their respectiv e Ford circles
(44) h 1 D h.p 1 ; b /; h 2 D h.p 2 ; a/; h 3 D h.1; 0/:
Such integ ers p 1 , p 2 e xist because the numbers a , b , c of a Mark o v triple are
pair wise copr ime. Moreo v er , this implies that the fractions in ( 43 ) are reduced, and
v 1 and v 2 are deter mined up to addition of a common integ er . By Proposition 6.2 ,
this decorated ideal tr iangle has edg e w eights
(45) a 1 D a ; a 2 D b ; a 3 D c
(see Fig. 14 f or notation).
Con v ersel y , e v er y ideal tr iangle Q v 1 Q v 2 Q v 3 with Q v 3 D 1 and rational Q v 1 , Q v 2 ,
that is decorated with the respectiv e Ford circles, has w eights ( 45 ) , and satisfies
Q v 1 < Q v 2 is obtained this w a y . (T o get the triangles with Q v 1 > Q v 2 , chang e c to  c
in equation ( 8 ) .) This implies that an y lift of a tr iangle of T to the h yperbolic
plane is GL 2 . Z / -related to v 1 v 2 v 3 . Use Proposition 13.4 with ı D 3 to deduce
that g is GL 2 . Z / -related to the g eodesic ending in x 0 ˙ r .

366 B. Springb orn
“ (b) ) (d) ”: Let O
T be the lift of the tr iangulation T to H 2 . The g eodesic g
crosses an infinite strip of tr iangles of O
T . By Proposition 13.4 , the signed distance
of g and an y Ford circle centered at a v er te x incident with this str ip is  log r .
W e claim that the signed distance to an y other Ford circle is lar g er . T o see this,
consider a v er te x v 2 Q [ ¹1º that is not incident with the tr iangle s tr ip, and
let  be a g eodesic ra y from v to a point p 2 g . Note that the projected ra y
 . / intersects  .g / at leas t once bef ore it ends in  .p / , and that the signed
distance to the firs t intersection is at least  log r .
“ (b) ^ (d) ) (c) ”: This f ollo ws directl y from r D q 9
4  1
c 2 < 3
2 .
“ (c) ) (a) ”: W e will sho w the contrapositiv e: If the g eodesic g does not project
to a simple closed g eodesic, then there is a F ord circle with signed distance
smaller than  log 3
2 C  , f or e v er y  > 0 .
There is nothing to sho w if at least one end of g is in Q [ ¹1º because
then the F ord circle at this end has signed distance 1 . So assume g does not
project to a simple closed g eodesic and both ends of g are irrational.
W e will recursiv el y define a sequence .T n / n  0 of ideal tr iangulations of the
modular tor us, with edg es labeled a n , b n , c n , suc h that the f ollo wing holds:
(1) The g eodesic  .g / has at leas t one pair of consecutiv e intersections with
the edg es a n , b n .
(2) The edg e w eights, which w e also denote b y a n , b n , c n , satisfy
a n  b n  c n ;
so that .a n ; b n ; c n / is a sor ted Mark o v triple.
(3) c n C 1 > c n
This pro v es the claim, because Propositions 13.2 and 13.4 impl y that f or eac h n ,
there is a horocy cle with signed distance to g less than  1
2 log  9
4  1
c 2
n  ; which
tends to  log 3
2 from abo v e as n ! 1 .
T o define the sequence .T n / , let T 0 be the tr iangulation with edge
w eights .1; 1; 1/ , with edg es labeled so that (1) holds.
Suppose the tr iangulation T n with labeled edg es is already defined f or some
n  0 . Define the labeled tr iangulation T n C 1 as f ollo ws. Since  .g / is not a
simple closed g eodesic, it intersects all three edg es. Because g has an ir rational
end (in f act, both ends are assumed to be irrational), there are infinitely man y
edg e intersections. Hence, there is pair of intersections with a n and b n ne xt to
an intersection with c n . If the sequence of intersections is a n b n c n , let T n C 1 be
the tr iangulation with edg es
.a n C 1 ; b n C 1 ; c n C 1 / D .a n ; c n ; b 0
n /;

The h yperbolic g eometr y of Mark o v’ s theorem 367
and if the sequence is b n a n c n , let T n C 1 be the tr iangulation with
.a n C 1 ; b n C 1 ; c n C 1 / D .b n ; c n ; a 0
n /;
where a 0
n and b 0
n are the ideal arcs obtained b y flipping the edges a n or b n in
T n , respectiv ely . By induction on n , one sees that (1), (2), (3) are satisfied f or
all n  0 .
“ (a) ^ (b) ) (e) ”: The Mark o v tr iples .a ; b ; c / v er ifying (b) are precisel y the
tr iples of edg e w eights of ideal tr iangulations containing the ideal arc c not
intersecting  .g / . The tr iangulations containing the ideal arc c f or m a doubly
infinite sequence in whic h neighbors are related b y a single edg e flip fixing c . In
this sequence, there is a uniq ue tr iangulation f or which the w eight c is lar g es t.
Proposition 15.2. Let g be a comple t e g eodesic in the hyperbolic plane, and let
X  R n Q be the se t of ends of lifts of simple closed g eodesics in the modular
torus. Then the f ollowing tw o stat ements ar e equiv alent :
(i) The ends of g ar e contained in Q [ ¹1º [ X .
(ii) F or some M >  log 3
2 ther e ar e only finit ely many (possibly zer o) F or d
cir cles h wit h signed distance d .g; h/ < M .
Pr oof. “ (i) ) (ii) ”: Consider the ends x k of g , k 2 ¹ 1; 2 º .
If x k 2 Q [ ¹1º , then g contains a ra y  k that is contained inside the F ord
circle at x k . In this case, let M k D 0 .
If x k 2 X , then x k is also the end of a g eodesic Q g that projects to a simple
closed g eodesic in the modular torus. By Proposition 15.1 , inf d .h; Q g / >  log 3
2 ,
where the infimum is tak en o v er all Ford circles h . Since g and Q g conv erg e
at x k , there is a constant M k >  log 3
2 and a ra y  k contained in g and ending
in x k such that d .h;  k />M k f or all Ford circles h .
The par t of g not contained in  1 or  2 is empty or of finite length, so it
can intersect the inter iors of at mos t finitel y man y Ford circles. This implies (ii)
with M D min .M 1 ; M 2 / .
“ (ii) ) (i) ”: T o sho w the contrapositiv e, assume (i) is false: A t least one end
of g is ir rational but not the end of a lift of a simple closed g eodesic in the
modular tor us. This implies that the projection  .g / intersects e v er y ideal arc in
the modular tor us infinitel y man y times. A dapt the ar gument f or the implication
“ (c) ) (a) ” in the proof of Proposition 15.1 to sho w that there is a sequence of
horocy cles .h n / and an increasing sequence of Mark o v numbers .c n / such that
d .g ; h n / <  1
2 log  9
4  1
c 2
n  . This implies that (ii) is f alse.

368 B. Springb orn
16. Dictionary : P oint – definite f orm.
Spectrum, classification of definite f orms,
and the F are y tessellation re visited
This section is about the h yperbolic geometry of definite binar y quadratic f or ms.
Its pur pose is to complete the dictionar y and pro vide a broader perspectiv e. This
section is not needed f or the proof of Mark o v’ s theorem.
If the binar y quadratic f or m ( 11 ) with real coefficients is positiv e or negativ e
definite, then the pol ynomial f .x ; 1/ has tw o comple x conjug ate roots. Let z .f /
denote the root in the upper half-plane, i.e.,
z .f / D  B C i p det f
A :
This defines a map f 7! z .f / from the space of definite f or ms to the h yperbolic
plane H 2 . It is surjectiv e and man y-to-one (an y non-zero multiple of a f or m is
mapped to the same point) and equiv ar iant with respect to the left GL 2 . R / -actions.
The signed distance of a horocy cle and a point in the h yperbolic plane is
defined in the ob vious w a y (positiv e f or points outside, neg ativ e f or points inside
the horocy cle). One obtains the f ollo wing proposition in the same w a y as the
cor responding statement about g eodesics (Proposition 10.1 ):
Proposition 16.1. The signed distance of t he hor ocycle h.p ; q / and t he point
z .f / 2 H 2 is
(46) d  h.p ; q /; z .f /  D log j f .p ; q / j
p det f :
This pro vides a geometric e xplanation f or the different beha vior of definite
binar y quadratic f or ms with respect to their minima on Z 2 :
F or all definite f or ms f , the infimum ( 15 ) is attained f or some .p ; q / 2 Z 2
and satisfies M .f /  2
p 3 . All f or ms equiv alent to p 2  p q C q 2 , and only those,
satisfy M .f / D 2
p 3 . But f or e v er y positiv e number m < 2
p 3 , there are infinitel y
man y equiv alence classes of definite f or ms with M .f / D m .
Algor ithms to deter mine the minimum M .f / of a definite q uadratic f or m f
are based on the reduction theor y f or quadratic f or ms. (The theor y of equiv alence
and reduction of binar y quadratic f or ms is usuall y de v eloped f or integ er f or ms, but
much of it carr ies o v er to f or ms with real coefficients.) The reduction algor ithm
descr ibed b y Con w a y [ 15 ] has a par ticularl y nice geometric inter pretation based
on the f ollo wing obser vation:
F or a point in the h yperbolic plane, the three nearest F ord circles (in the
sense of signed dis tance) are the F ord circles at the v er tices of the F are y tr iangle

The h yperbolic g eometr y of Mark o v’ s theorem 369
containing the point. (If the point lies on an edg e of the F arey tessellation, the
third nearest F ord circle is not unique.)
A ckno wledg ements. I w ould lik e to thank Oliv er Pretzel, who ga v e me a first
glimpse of this subject some 25 y ears ago, and Ale x ander V eselo v , who made
me look ag ain. Las t but not least, I w ould like to thank the anon ymous ref erees
f or their insightful comments.
This research w as suppor ted b y DFG SFB/TR 109 “Discretization in Geometr y
and Dynamics ”.
R ef erences
[ 1 ] R. Abe and I. R. Aitchison , Geometr y and Mark off’ s spectrum f or Q .i / , I. T rans.
Amer . Math. Soc. 365 (2013), 6065–6102. Zbl 1301.57011 MR 3091276
[ 2 ] M. Aigner , Marko v’ s theor em and 100 year s of t he uniqueness conjectur e . Spr inger ,
Cham (2013). Zbl 1301.57011 MR 3091276
[ 3 ] V . I. Arnold , Higher -dimensional continued fractions. Regul. Chao tic Dyn. 3 (1998),
10–17. MR 1704965
[ 4 ] T sepnye dr obi (Continued fr actions, in Russian) . MT sNMO, Mosco w (2001).
[ 5 ] A. F . Beardon , J. Lehner , and M. Shein gorn , Closed g eodesics on a Riemann
sur f ace with application to the Mark o v spectr um. T r ans. Amer . Math. Soc. 295
(1986), 635–647. Zbl 0597.10024 MR 0833700
[ 6 ] E. B ombieri , Continued fractions and the Mark off tree. Expo. Math. 25 (2007),
187–213. Zbl 1153.11030 MR 2345177
[ 7 ] F . Bon ahon , Lo w-dimensional g eometr y , v ol. 49 of S tudent Mathematical Libr ar y .
Amer ican Mathematical Society , Pro vidence, RI; Institute f or A dv anced S tudy
(IAS), Pr inceton, NJ (2009). Zbl 1176.57001 MR 2522946
[ 8 ] B. H. Bo wditch , A proof of McShane’ s identity via Mark off tr iples. Bull. London
Math. Soc. 28 (1996), 73–78. Zbl 0854.57009 MR 1356829
[ 9 ] Mark off triples and quasi-Fuchsian groups. Pr oc. London Mat h. Soc. (3) 77
(1998), 697–736. Zbl 0928.11030 MR 1643429
[ 10 ] J. O. Button , The uniqueness of the prime Markoff numbers. J. London Mat h. Soc.
(2) 58 (1998), 9–17. Zbl 0932.11020 MR 1666058
[ 11 ] J. W . S. C assels , An intr oduction t o Diophantine appr o ximation . Cambr idge T racts
in Mathematics and Mathematical Ph y sics, No. 45. Cambridge U niv ersity Press,
Ne w Y ork (1957). Zbl 0077.04801 MR 0349591
[ 12 ] H. Cohn , Approac h to Mark off’ s minimal f or ms through modular functions. Ann.
of Math. (2) 61 (1955), 1–12. Zbl 0064.04303 MR 0067935
[ 13 ] R epresentation of Mark off’ s binary quadratic f or ms b y g eodesics on a perf o-
rated tor us. A cta Arith. 18 (1971), 125–136. Zbl 0218.10041 MR 0288079

370 B. Springb orn
[ 14 ] Mark off f or ms and primitiv e w ords. Math. Ann. 196 (1972), 8–22. Zbl 0227.10018
MR 0297847
[ 15 ] J. H. Conw a y , The sensual (quadr atic) form , v ol. 26 of Car us Mathematical
Monog r aphs . Mathematical Association of America, W ashington, DC (1997).
Zbl 1310.11001 MR 1478672
[ 16 ] J. H. Conw a y and R. K. Guy , The book of numbers . Copernicus, Ne w Y ork (1996).
Zbl 0866.00001 MR 1411676
[ 17 ] D. J. Crisp , The Markoff spectrum and g eodesics on the punctur ed torus . PhD
thesis, U niv ersity of A delaide (1993).
[ 18 ] D. Crisp , S. Dziadosz , D. J. Garity , T . Insel , T . A. Schmidt and P . Wiles ,
Closed cur v es and geodesics with tw o self-intersections on the punctured
tor us. Monatsh. Math. 125 (1998), 189–209. Zbl 0901.57022 MR 1614271
[ 19 ] D. J. Crisp and W . Moran , Single self-intersection g eodesics and the Mark off
spectr um. In N umber theory with an emphasis on the Mar koff spectrum (Pro vo,
UT, 1991) , v ol. 147 of Lecture N ot es in Pur e and Appl. Mat h. Dekk er , Ne w
Y ork (1993). pp. 83–93. Zbl 0803.11029 MR 1219327
[ 20 ] T . W . Cusick and M. E. Flahive , The Markoff and Lag r ang e spectr a , v ol. 30
of Mathematical Sur v eys and Monog r aphs . American Mathematical Society ,
Pro vidence, RI (1989). Zbl 0685.10023 MR 1010419
[ 21 ] S. G. D ani and A. Nogueira , Continued fractions f or comple x numbers and v alues
of binar y quadratic f or ms. T rans. Amer . Math. Soc. 366 (2014), 3553–3583.
Zbl 06303170 MR 3192607
[ 22 ] B. N . Delone , The St. Pet er sbur g sc hool of number theor y , v ol. 26 of Hist or y
of Mathematics . American Mathematical Society , Pro vidence, RI (2005).
T ranslated from the 1947 R ussian original. Zbl 1074.11002 MR 0091894
[ 23 ] G. L. Dirichlet , V erallg emeiner ung eines Satzes aus der Lehre v on den Ketten-
br üc hen nebst einig en An w endung en auf die Theor ie der Zahlen. Beric ht über
die zur Bekanntmac hung g eeignet en V erhandlung en der König lic h Pr eußischen
Akademie der Wissensc haf ten zu Ber lin (1842), 93–95.
[ 24 ] V . V . Fock and A. B. Gonchar o v , Dual Teichmüller and lamination spaces. In
A. P ap adopoulos (ed.), Handbook of Teic hmüller theory. Vol. I , v ol. 11 of
IRMA Lect. Math. Theor . Phys. Eur . Math. Soc., Zür ich (2007). pp. 647–684.
Zbl 1162.32009 MR 2349682
[ 25 ] L. R. Ford , Rational appro ximations to ir rational comple x numbers. T r ans. Amer .
Math. Soc. 19 (1918), 1–42. JFM 46.0275.04 MR 1501085
[ 26 ] On the closeness of approach of comple x rational fractions to a comple x
ir rational number . T r ans. Amer . Math. Soc. 27 (1925), 146–154. JFM 51.0157.03
MR 1501304
[ 27 ] Fractions. Amer . Math. Mont hly 45 (1938), 586–601. JFM 64.0148.03 MR 1524411
[ 28 ] G. Fr obenius , Über die Mark offschen Zahlen. Sitzungsberic hte der König lic h
Pr eussisc hen Ak ademie der W issensc haf ten zu Ber lin (1913), 458–487. R eprinted
in: F . G. Frobenius. Gesammelte Abhandlung en , v olume III. Spr ing er - V erlag,
Ber lin–Ne w Y ork, 1968, pages 598–627. JFM 44.0255.01

The h yperbolic g eometr y of Mark o v’ s theorem 371
[ 29 ] D. Fuchs and S. T ab a chnik o v , Mathematical omnibus . Amer ican Mathematical
Society , Pro vidence, RI (2007). Zbl 1318.00004 MR 2350979
[ 30 ] D. S. Gorshk o v , Geometry of Lobache v skii in connection with cer tain ques tions
of ar ithmetic (Russian). Zap. N auc hn. Semin. Leningr . Otd. Mat. Inst. St eklo v a
76 (1977), 39–85. MR 0563093 . English translation in J. So vie t Math. 16 (1981)
788–820. Zbl 0453.10032
[ 31 ] A. Haas , Diophantine appro ximation on hyperbolic Riemann surf aces. A cta Math.
156 (1986), 33–82. Zbl 0593.10028 MR 0822330
[ 32 ] G. H. Hard y and E. M. Wright , An intr oduction to t he theory of numbers . Oxf ord
U niv ersity Press, Oxf ord, sixth ed. (2008). Re vised by D. R. Heath-Bro wn
and J. H. Sil v er man, with a f ore w ord by Andre w Wiles.
[ 33 ] A. Ha tcher , T opology of numbers. Book in preparation, https://www.math.
cornell.edu/~hatcher/TN/TNpage.html (accessed 2017-02-07).
[ 34 ] On tr iangulations of surfaces. T opology Appl. 40 (1991), 189–194. Zbl 0727.57012
MR 1123262
[ 35 ] C. Hermite , Sur l’introduction des v ariables continues dans la théor ie des nombres.
J. Reine Ang ew . Math. 41 (1851), 191–216. ERAM 041.1126cj MR 1578717
[ 36 ] S. Hersonsky and F . P a ulin , Diophantine appro ximation f or negativ el y curv ed
manif olds. Math. Z. 241 (2002), 181–226. Zbl 1037.53020
[ 37 ] J. H. Hubb ard , The KAM theorem. In Charpentier , Lesne and Nik olski (eds.),
K olmogor o v’ s Heritag e in Mathematics . Springer , Berlin (2007). pp. 215–238.
Zbl 1369.37070
[ 38 ] A. Hur witz , Ueber die ang enäher te Darstellung der Irrationalzahlen durch rationale
Brüche. Mat h. Ann. 39 (1891), 279–284. JFM 23.0222.02 MR 1510702
[ 39 ] Ueber die Reduction der binären quadratisc hen Formen. Math. Ann. 45 (1894),
85–117. JFM 25.0313.01 MR 1510855
[ 40 ] F . Klein , V orlesung en über die Theorie der elliptisc hen Modulfunctionen. Ausg ear -
beite t und v er volls tändigt v on Robert Fric ke , v ol. 1. T eubner , Leipzig (1890).
[ 41 ] Ueber eine g eometrische Auffassung der g e wöhnlichen Kettenbr uc hentwic k e-
lung. N ac hric hten v on der Gesellsc haf t der Wissensc haf ten zu Götting en.
Mathematisc h-Physikalisc he Klasse 1895 (1895), 357–359. JFM 26.0229.02
[ 42 ] A usg ewählte Kapitel der Zahlentheorie I. V orlesung, g ehalten im Win-
tersemester 1895/96. A usg earbeitet v on A. Sommer f eld. Götting en (1896).
JFM 27.0163.02
[ 43 ] A. K orkine and G. Zolot areff , Sur les f or mes quadratiq ues. Math. Ann. 6 (1873),
366–389. JFM 05.0109.01 MR 1509828
[ 44 ] M. L. Lang and S. P . T an , A simple proof of the Mark off conjecture f or pr ime
po wers. Geom. Dedicata 129 (2007), 15–22. Zbl 1133.11023 MR 2353978
[ 45 ] A.-M. Legendre , Théorie des nombr es , v ol. 1. Firmin-Didot, Paris (1830).
[ 46 ] J. Lehner and M. Sheingorn , Simple closed g eodesics on H C =  .3/ ar ise from
the Mark o v spectr um. Bull. Amer . Math. Soc. (N .S.) 11 (1984), 359–362.
Zbl 0556.10015 MR 0752798

372 B. Springb orn
[ 47 ] A. V . Mal yšev , Mark o v and Lag rang e spectra (sur v e y of the literature). (Russian).
Zap. N aučn. Sem. Lening r ad. Otdel. Mat. Inst. S t eklo v . (LOMI) 67 (1977), 5–38,
225. Enlish translation in J. So viet Mat h. 16 (1981) 767–788. MR 044187 6
[ 48 ] A. Mark off , Sur les f ormes quadratiques binaires indéfinies. Math. Ann. 15 (1879),
381–406. JFM 11.0147.01
[ 49 ] Sur les f or mes quadratiques binaires indéfinies. (Sécond mémoire). Mat h. Ann.
17 (1880), 379–399. JFM 12.0143.02 MR 1510073
[ 50 ] G. McShane , A r emar kable identity f or lengths of curves . PhD thesis, U niv ersity
of W ar wick, Mathematics Ins titute (1991). http://wrap.warwick.ac.uk/id/
eprint/4008 .
[ 51 ] G. McShane and H. P arlier , Multiplicities of simple closed geodesics and
h ypersurfaces in Teic hmüller space. Geom. T opol. 12 (2008), 1883–1919.
MR 3389436
[ 52 ] G. McShane and I. Rivin , Simple cur v es on h yperbolic tor i. C. R. A cad. Sci. P aris
Sér . I Math. 320 (1995), 1523–1528. Zbl 0835.53050 MR 1340065
[ 53 ] M. Mirzakhani , Gro wth of the number of simple closed geodesics on h yperbolic
sur f aces. Ann. of Mat h. (2) 168 (2008), 97–125. Zbl 1177.37036 MR 2415399
[ 54 ] L. Mosher , T iling the projectiv e f oliation space of a punctured surface. T r ans.
Amer . Math. Soc. 306 (1988), 1–70. Zbl 0647.57005 MR 0927683
[ 55 ] R. C. Penner , The decorated Teichmüller space of punctured surf aces. Comm. Math.
Phy s. 113 (1987), 299–339. Zbl 0642.32012 MR 0919235
[ 56 ] Decor at ed T eic hmüller Theor y . QGM Master Class Series. European Mathe-
matical Society (EMS), Zür ic h (2012). Zbl 1243.30003 MR 3052157
[ 57 ] S. Perrine , From Frobenius to Riedel: anal y sis of the solutions of the Mark off
equation. https://hal.archives- ouvertes.fr/hal- 00406601 (2009).
[ 58 ] O. Perr on , Die Lehr e von den Ke tt enbrüc hen. Bd I. Elementar e Ke tt enbrüc he . B. G.
T eubner , Stuttg ar t, 3rd ed. (1954). Zbl 0056.05901 MR 0064172
[ 59 ] N . Riedel , On the mark off eq uation. arXiv :1208.4032 [math.NT] (2012).
[ 60 ] K. F . R oth , Rational appro ximations to alg ebraic numbers. Mathematika 2 (1955),
1–20; cor r ig endum, 168. Zbl 0064.28501 MR 0072182
[ 61 ] A. L. Schmidt , Diophantine appro ximation of comple x numbers. Acta Mat h. 134
(1975), 1–85. Zbl 0329.10023 MR 0422168
[ 62 ] Minimum of quadratic f or ms with respect to Fuchsian groups. I. J. R eine
Ang ew . Math. 286/287 (1976), 341–368. Zbl 0332.10015 MR 0457358
[ 63 ] Minimum of quadratic f or ms with respect to Fuc hsian groups. II. J. Reine
Ang ew . Math. 292 (1977), 109–114. Zbl 0347.10017 MR 0457359
[ 64 ] P . Schmutz , Sy stoles of ar ithmetic surfaces and the Mark off spectr um. Math. Ann.
305 (1996), 191–203. Zbl 0853.11054 MR 1386112
[ 65 ] P . Schmutz Schaller , Geometry of Riemann sur f aces based on closed g eodesics.
Bull. Amer . Math. Soc. (N .S.) 35 (1998), 193–214. Zbl 1012.30029 MR 1609636
[ 66 ] C. Series , The g eometry of Markoff numbers. Mat h. Int ellig encer 7 (1985), 20–29.
Zbl 0566.10024 MR 0795536

The h yperbolic g eometr y of Mark o v’ s theorem 373
[ 67 ] The modular sur f ace and continued fractions. J. London Mat h. Soc. (2) 31
(1985), 69–80. Zbl 0545.30001 MR 0810563
[ 68 ] K. Sp alding and A. P . Veselo v , L y apuno v spectr um of Mark o v and Euclid trees.
N onlinearity 30 (2017), 4428–4453. Zbl 06822780 MR 3734142
[ 69 ] Gro wth of values of binary quadratic f or ms and Con w a y r iv ers. Preprint (2017).
[ 70 ] A. Speiser , Eine g eometrische Figur zur Zahlentheor ie. A ct es de la Société
Helvétiq ue des Sciences N atur elles 104 (1923), 113–114.
[ 71 ] D. Sulliv an , Disjoint spheres, appro ximation b y imaginar y quadratic numbers,
and the log ar ithm la w f or g eodesics. A cta Math. 149 (1982), 215–237.
Zbl 0517.58028 MR 0688349
[ 72 ] K. T . V ahlen , Über Näher ungsw er te und Kettenbrüche. J. R eine Ang ew . Math. 115
(1895), 221–233. JFM 26.0230.01 MR 1580401
[ 73 ] L. Y . Vulakh , F are y polytopes and continued fractions associated with dis-
crete h yperbolic groups. T r ans. Amer . Math. Soc. 351 (1999), 2295–2323.
Zbl 0924.11061 MR 1467 477
[ 74 ] D. Za gier , On the number of Markoff numbers belo w a giv en bound. Math. Comp.
39 (1982), 709–723. Zbl 0501.10015 MR 0669663
[ 75 ] J. Züllig , Geometrisc he Deutung unendlic her K e ttenbrüc he und ihre Appr o xima-
tionen dur c h rationale Zahlen . Orell-Füssli, Zürich (1928). JFM 54.0210.02
( Reçu le 11 mar s 2017 )
Bor is Sprin gborn , T ec hnisc he U niv ersität Berlin, Ins titut für Mathematik, MA 8-3,
S traße des 17. Juni 136, 10623 Ber lin, Ger man y
e-mail: bor is.springbor [email protected]
© F ondation L ’Enseignement Ma théma tique

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