Lattice point asymptotics and volume growth on Teichmüller space

LATTICE POINT ASYMPTOTICS AND VOLUMEGROWTH ON TEICHMÜLLER SPACE

JAYADEV ATHREYA, ALEXANDER BUFETOV, ALEX ESKIN, and MARYAMMIRZAKHANI

AbstractWe apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space.Let X be a point in Teichmüller space, and let BR.X/ be the ball of radius R cen-tered at X (with distances measured in the Teichmüller metric). We obtain asymptoticformulas as R tends to infinity for the volume of BR.X/, and also for the cardinalityof the intersection of BR.X/ with an orbit of the mapping class group.

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10552. Outline of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10583. The Hodge norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10704. The multiple zero locus . . . . . . . . . . . . . . . . . . . . . . . . . 10905. Volume asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095Appendix. Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . 1102References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109

1. IntroductionMuch of the study of the geometry and dynamics on Teichmüller and moduli spacesis inspired by analogies with negatively curved spaces. Two classical problems in

DUKE MATHEMATICAL JOURNALVol. 161, No. 6, © 2012 DOI 10.1215/00127094-1548443Received 28 February 2010. Revision received 17 September 2011.2010 Mathematics Subject Classification. Primary 37A25; Secondary 30F60.Athreya’s work partially supported by National Science Foundation grants DMS-0603636, DMS-0244542, and

DMS-1069153.Bufetov’s work partially supported by an Alfred P. Sloan Research Fellowship, a Dynasty Foundation Fellow-

ship, by President of the Russian Federation grants MK-4893.2010.1 and MK-6734.2012.1, by the Programon Dynamical Systems and Mathematical Control Theory of the Presidium of the Russian Academy of Sci-ences, by RFBR-CNRS grant 10-01-93115, and by RFBR grant 11-01-00654.

Eskin’s work partially supported by National Science Foundation grants DMS-0244542, DMS-0604251, andDMS-0905912.

Mirzakhani’s work partially supported by the Clay Foundation and by National Science Foundation grant DMS-0804136.

1055

http://dx.doi.org/10.1215/00127094-1548443

1056 ATHREYA, BUFETOV, ESKIN, and MIRZAKHANI

the negative curvature setting are the questions of lattice point counting and volumegrowth entropy. Let M be a compact negatively curved Riemannian manifold, andlet fM be its universal cover. The fundamental group … D �1.M/ acts on fM byisometries. Let mBM denote the Bowen–Margulis measure on fM . Given that X;Y 2fM , R > 0, let BR.X/� fM be the ball of radius R centered at X . Let p W fM !M

be the natural projection.In his Ph.D. dissertation, Margulis showed the following.

THEOREM 1.1 ([Mar, Theorems 8, 9])There is a function c WM �M !R

C so that, for every X;Y 2fM ,

j… � Y \BR.X/j � c�p.X/;p.Y /

�ehR; (1.1)

mBM�BR.X/

��ZM

c�p.X/; z

�dmBM.z/

�ehR; (1.2)

where h > 0 is the topological entropy of the geodesic flow. Here and below, thenotation A�B means that A=B! 1 as R!1.

In this paper, we apply some of the ideas of Margulis from [Mar] to the prob-lems of counting lattice points and understanding volume growth entropy in Teich-müller space. Our main results (Theorems 1.2, 1.3) are analogues of Theorem 1.1for Teichmüller space and the fact that the Bowen–Margulis measure coincides withthe Lebesgue measure (see Section 2.2). In the rest of this introduction, we give therequired background and definitions of Teichmüller space and quadratic differentialsand state our main results.

1.1. Teichmüller space and quadratic differentialsWe briefly recall some definitions. (For a more detailed introduction, see, e.g.,[FM].) Let g � 2, and let †g be a compact topological surface of genus g. Let Mg

and Tg denote the moduli space and the Teichmüller space of compact Riemann sur-faces of genus g.

That is, Tg is the space of equivalence classes of pairs .X;f /, where X is a com-pact genus g Riemann surface and f W †g ! X is a diffeomorphism (known as amarking). The equivalence relation is given by .X;f /� .Y;h/ if there is a biholo-morphism � WX! Y so that h�1 ı � ı f is isotopic to the identity.

Let � be the mapping class group of †g given by isotopy classes of orientation-preserving diffeomorphisms of †g . That is, � DDiffC.†g/=DiffC0 .†g/. The group� acts on Tg by changing the marking, and we have Mg D Tg=� .

Here Tg carries a natural Finsler metric invariant under � known as the Teich-müller metric. It is given by measuring the quasi-conformal dilatation between sur-

LATTICE POINT ASYMPTOTICS AND VOLUME GROWTH 1057

faces (see, e.g., [FM]). The cotangent space of Tg at a point X can be identifiedwith the vector space Q.X/ of holomorphic quadratic differentials on X: Recall that,givenX 2 Tg , a quadratic differential q 2Q.X/ is a tensor locally given by �.z/dz2,where � is holomorphic with respect to the complex structure given by X . Then thespace QTg D ¹.q;X/ j X 2 Tg ; q 2Q.X/º is the cotangent space of Tg : In this set-ting, the Teichmüller metric corresponds to the norm

kqkT D

ZX

j�.z/jjdzj2

on QTg :

Let Q1Mg (resp., Q1Tg ) denote the bundle of unit-norm holomorphic quadraticdifferentials on Mg (resp., Tg ). We have Q1Mg DQ1Tg=� .

The space Q1Tg carries a natural smooth measure � preserved by the action of� and such that �.Q1Mg/ <1 (see [Ma2], [V1]). We will fix a normalization for �in Section 2.2.

We have a natural projection

� WQ1Tg ! Tg ;

and we set mD ��.

1.2. Statement of resultsLet X;Y 2 Tg be arbitrary points, and let BR.X/ be the ball of radius R in the Teich-müller space in the Teichmüller metric, centered at the point X .

Our main results are the following two theorems.

THEOREM 1.2 (lattice point asymptotics)As R!1,

j� � Y \BR.X/j �1

hm.Mg/ƒ.X/ƒ.Y /ehR;

where h D 6g � 6 is the entropy of the Teichmüller geodesic flow with respect toLebesgue measure (see [V2]) and where ƒ is a bounded function called theHubbard–Masur function, which we define in Section 2.3.

THEOREM 1.3 (volume asymptotics)As R!1,

m�BR.X/

��

1

hm.Mg/ehRƒ.X/ �

ZMg

ƒ.Y /dm.Y /:


We also prove versions of the above theorems in sectors (see Theorems 2.9, 2.10and 5.2 below).

Recent work by Dumas [Du] implies that the Hubbard–Masur function is in factconstant.

1.3. Organization of the paperThis paper is organized as follows. In Section 2, we describe the main ingredientsin the proofs of our main theorems. We construct foliations and measures satisfyingthe Margulis property of uniform expansion in Section 2.2. We define the Hubbard–Masur function ƒ and describe its basic properties and relation to counting multi-curves in Section 2.3. In Section 2.4, we state a crucial lemma, Proposition 2.5, onmeasure in polar coordinates (we postpone the proof to Section 3.2). In Section 2.5,we show how to use mixing to obtain equidistribution results, and we apply themto the counting problem in Section 2.6. In Section 3, we recall the definition of theHodge norm for abelian differentials (Section 3.1), and we extend and modify it toquadratic differentials (Section 3.3). We use this to define a distance along leaves ofthe stable and unstable foliations for the Teichmüller geodesic flow, and in Section 3.5we compare it to other distances on Tg . We prove a nonexpansion result for thisdistance, Theorem 3.15, in Section 3.6. In Section 4, we use the results of Section 3to prove the key estimates Theorem 2.6 and Theorem 2.7 from Section 2.5. Finally,in Section 5, we prove our volume asymptotics result, Theorem 1.3, and relate it tocounting multicurves.

2. Outline of proof

2.1. Notation and backgroundTeichmüller geodesic flow. We recall that when g > 1; the Teichmüller metric is notRiemannian. However, geodesics in this metric are well understood. A quadratic dif-ferential q 2 QTg with zeros at x1; : : : ; xk is determined by an atlas of charts ¹�iºmapping open subsets of†g �¹x1; : : : ; xkº to R

2 such that the change of coordinatesare of the form v!˙vC c: Therefore, the group SL2.R/ acts naturally on QMg byacting on the corresponding atlas; given A 2 SL2.R/, A � q 2QMg is determined bythe new atlas ¹A�iº:

Let gt D Œ et 00 e�t

�. The action of the diagonal subgroup ¹gt j t 2 Rº is the Teich-müller geodesic flow for the Teichmüller metric (see [FM]).

Warning. In our normalization for the Teichmüller metric, the Teichmüller dis-tance between �.gtq/ and �.q/ is t . This normalization (and thus our value for theentropy h) differs by a factor of 2 from, for example, that of [V2]. Our normalizationis chosen in such a way that the top Lyapunov exponent of the Kontsevich–Zorich


cocycle is equal to 1. In this case, the top Lyapunov exponent of the flow is equalto 2. (For a detailed discussion of the connection between the Lyapunov exponentsof the Teichmüller geodesic flow and the Kontsevich–Zorich cocycle, see, e.g., [Fo],where the same speed normalization is used for the Teichmüller geodesic flow as inour paper.)

We have the following.

THEOREM 2.1 (see Veech [V1], Masur [Ma2])The space Q1Mg carries a unique (up to normalization) measure � in the Lebesguemeasure class such that� �.Q1Mg/ <1;� the action of SL2.R/ is volume-preserving and ergodic;� the Teichmüller geodesic flow is mixing.

Extremal lengths. Let X be a Riemann surface. Then the extremal length of asimple closed curve � on X is defined by

Ext� .X/D sup�

`� .�/2

Area.X;�/;

where the supremum is taken over all metrics � conformally equivalent to X andwhere `� .�/ denotes the length of � in the metric �. The extremal length can beextended continuously from the space of simple closed curves to the space MF ofmeasured foliations in such a way that Exttˇ .X/ D t2Extˇ .X/ (see [Ke]). On theother hand, by the uniformization theorem, each point X 2 Tg has a complete hyper-bolic metric �0 of constant curvature �1 in its conformal class. In general, for anysimple closed curve ˛,

Ext˛.X/

`˛.X/1

2e`˛.X/=2; (2.1)

where `˛.X/ is the length of the geodesic representative of ˛ on X with respect tothe hyperbolic metric �0 (see [M]). Also, given X , there exists a constant CX suchthat

1

CX`˛.X/

pExt˛.X/ CX`˛.X/:

The following result (see [Ke]) relates the ratios of extremal lengths to the Teich-müller distance.

THEOREM 2.10 ([Ke, Theorem 4])Given X;Y 2 Tg , the Teichmüller distance between X and Y is given by


dT .X;Y /D supˇ2C

log�pExtˇ .X/p

Extˇ .Y /

�;

where C is the set of simple closed curves on †g :

2.2. The Margulis propertyLet gt be the Teichmüller geodesic flow on Q1Tg . Note that gt commutes with theaction of � and preserves the measure �. We will be using the mixing property of thedynamical system .gt ;Q

1Mg ;�/ in Section 2.5.Recall that a quadratic differential q is uniquely determined by its imaginary and

real measured foliations �.q/ given by Im.q1=2/ and C.q/ given by Re.q1=2/. Inthis notation, we have gtq D gt .C.q/; �.q// D .etC.q/; e�t�.q//. (See, e.g.,[FLP] for more details on measured foliations.)

The flow gt preserves the following foliations:(1) F ss , whose leaves are sets of the form ¹q W C.q/D constº;(2) F uu, whose leaves are sets of the form ¹q W �.q/D constº.

In other words, for q0 2Q1Tg , a leaf of F ss is given by

˛ss.q0/D®q 2Q1Tg W

C.q/D C.q0/¯;

and a leaf of F uu is given by

˛uu.q0/D®q 2Q1Tg W

�.q/D �.q0/¯:

Note that the foliations F ss , F uu are invariant under both gt and � ; in particular,they descend to the moduli space Q1Mg .

We also consider the foliations F u whose leaves are defined by

˛u.q/D[t2R

gt˛uu.q/

and F s whose leaves are defined by

˛s.q/D[t2R

gt˛ss.q/:

Denote by p WMF !P MF the natural projection from the space of measuredfoliations onto the space of projective measured foliations. Now we may write

˛s.q0/D®q 2Q1Tg W p

�C.q/

�D p

�C.q0/

�¯I

˛u.q0/D®q 2Q1Tg W p

��.q/

�D p

��.q0/

�¯:


The foliations F u and F ss form a complementary pair in the sense of Margulis[Mar] (so do F s and F uu, but the first pair will be more convenient for us). Note thatthe foliations F ss;F s;F u;F uu are, respectively, the strongly stable, stable, unsta-ble, and strongly unstable foliations for the Teichmüller flow in the sense of Veech[V2] and Forni [Fo]. (See Theorem 3.15 below for further results in this direction.)

Conditional measures. The main observation, lying at the center of our construc-tion, is the following. Each leaf ˛uu of the foliation F uu, as well as each leaf ˛ss

of the foliation F ss , carries a globally defined normalized conditional measure �˛ss ,invariant under the action of the mapping class group, and having, moreover, the fol-lowing property:(1) .gt /��˛uu D exp.�ht/�gt˛uu and(2) .gt /��˛ss D exp.ht/�gt˛ss ,where hD 6g � 6 is the entropy of the flow gt on Q1Mg with respect to the smoothmeasure �.

The measures �˛uu and �˛ss may be constructed as follows. Let denote theThurston measure on MF (see [FLP]). Note that each leaf ˛s of F s is homeomorphicto an open subset of MF via the map �. We can thus define the conditional measureon this leaf to be the pullback of , denoting this measure by �˛s . Similarly, one candefine the conditional measures �˛u on leaves ˛u of F u.

To define the measures on leaves of F ss (and F uu), we restrict the conditionalmeasure from a leaf of F s to a leaf of F ss (and similarly, from a leaf of F u to a leafof F uu). This can be done explicitly in the following way. For a subset E �MF , letCone.E/ denote the cone based at the origin and ending at E (i.e., the union of allthe line segments connecting the origin and points of E). We write N.E/ to denote.Cone.E//. Now for a set F � ˛ss , we define �˛ss .F /D N.�.F //. Similarly, forF � ˛uu, �˛uu.F / is defined to be N.C.F //.

In particular, if ˛1 and ˛2 are two leaves of the foliation F u and if U1 � ˛1 andU2 � ˛2 are chosen in such a way that C.U1/D C.U2/, then we have �˛1.U1/D�˛2.U2/. The equality C.U1/D C.U2/ is equivalent to the statement that U1 maybe taken to U2 by holonomy along the leaves of the strongly stable foliation F ss ; theequality �˛1.U1/D �˛2.U2/ thus means that the smooth measure � has the propertyof holonomy invariance with respect to the pair of foliations .F u;F ss/.

This construction allows us to apply the arguments of Margulis [Mar] and tocompute the asymptotics for the volume of a ball of growing radius in Teichmüllerspace, as well as the asymptotics of the number of elements in the intersection ofa ball with the orbit of the mapping class group. The approach is similar, as notedabove, to that of [EMc].

Normalization of �. For convenience, we normalize the measure � so that locallyd�D d�˛ud�˛ss D d�˛sd�˛uu .


2.3. The Hubbard–Masur functionThe Hubbard–Masur theorem. The Hubbard–Masur theorem (see [HMa, Main Theo-rem]) states that, given any point X 2 Tg and any measured foliation ˇ 2MF , thereexists a unique holomorphic quadratic differential q on X such that C.q/D ˇ. Wealso have the identity Area.q/D Extˇ .X/.

The measure sX and the multiple zero locus. For X 2 Tg , we consider the unit(co)tangent sphere S.X/D ¹q 2Q1Tg W �.q/D Xº at the point X . The conditionalmeasure of � on the sphere S.X/will be denoted by sX . It is by definition normalizedso that sX .S.X//D 1.

Let P .1; : : : ; 1/ � Q1Tg denote the subset where the all zeros of the quadraticdifferential are distinct. Here P .1; : : : ; 1/ is called the principal stratum. Its comple-ment in Q1Tg is the multiple zero locus. It is easy to see that the measures sX aredefined for all X and that this family of measures is smooth away from the multiplezero locus. We will also need the following.

THEOREM 2.2For any X 2 Tg , the measure sX gives zero weight to the multiple zero locus.

ProofSee the Appendix. �

The Hubbard–Masur function. Let q 2 Q1Tg , and let ˛u.q/ be the leaf of thefoliation F u containing q. By the Hubbard–Masur theorem, the projection � inducesa continuous bijection between ˛u.q/ and Tg which is smooth away from the mul-tiple zero locus. The mapping � thus takes the globally defined conditional measure�˛u.q/ on the fiber ˛u.q/ to a measure on the Teichmüller space; the resulting mea-sure ��.�˛u.q// is absolutely continuous with respect to the smooth measure m. Fur-thermore, by Theorem 2.2, the measure m is also absolutely continuous with respectto ��.�˛u.q//; indeed, away from the multiple zero locus the mapping � is smoothwith a smooth inverse, and the multiple zero locus has measure zero by Theorem 2.2.

We may therefore consider the corresponding Radon–Nikodym derivative. Intro-duce a function �C WQ1Tg!R by the formula

�C.q/Ddm

d.��.�˛u.q///

��.q/

�:

Similarly, we define �� WQ1Tg !R via the formula

��.q/Ddm

d.��.�˛s.q///

��.q/

�:


We set

ƒ.X/D

ZS.X/

�C.q/dsX .q/D

ZS.X/

��.q/dsX .q/:

The equality of the two integrals will be justified by Proposition 2.3. Note that thefunctions �C, ��, andƒ are �-invariant. We call �C, ��, andƒ the Hubbard–Masurfunction.

Note that by the Hubbard–Masur theorem, � (or C) defines a homeomorphismbetween the space of all quadratic differentials atX (with arbitrary area) and the spaceMF . This homeomorphism restricts to a homeomorphism between S.X/ and the set

EX D®ˇ 2MF W Extˇ .X/D 1

¯;

where Extˇ .X/ is the extremal length at X of the measured foliation ˇ. Let ı˙X WEX ! S.X/ denote the inverse of ˙.

It is easy to see that the functions �˙ are smooth on the complement of themultiple zero locus.

PROPOSITION 2.3 (properties of �C, ��, ƒ)Let X D �.q/. Then we have:

(i) �C.q/Dd.ı�

X/�. N�/

dsX.q/ and ��.q/D

d.ıC

X/�. N�/

dsX.q/;

(ii) ƒ.X/D N��.S.X//

�D N

�C.S.X//

�;

(iii) ƒ.X/D N.EX /D �¹ˇ 2MF W Extˇ .X/ 1º

�;

(iv) ƒ.X/D N�¹ˇ 2ML W Extˇ .X/ 1º

�, where ML is the space of measured

laminations.In (iv), by abuse of notation, denotes the Thurston measure on ML and N.E/D.Cone.E//.

ProofProperty (i) follows from the fact that d�D d�˛u d�˛ss , the fact that �� .˛

ss/D N,and the fact that if we write, for q 2 Q1Tg , q D .X; v/, where X D �.q/ and v 2S.X/, then d�.q/D dm.X/dsX .v/. Property (ii) follows from (i) after making thechange of variable q! �.q/ in the definition of ƒ. Property (iii) follows from thefact that the image �.S.X// consists exactly of EX . Finally, (iv) is easily seen to beequivalent to (iii). �

The following is proved in Section 5.

THEOREM 2.4 (boundedness of ƒ)There exists a constant M such that ƒ.X/M for all X 2 Tg .


(For another interpretation of ƒ in terms of the asymptotics of the number ofmulticurves, see (5.3) below.)

2.3.1. Conformal densitiesLet the measure X on P MF be the image of ��.q/dsX .q/ under the identificationof S.X/ and P MF . That is,

X .U /D �®ˇ̌Œ� 2 U;Ext�.X/ 1

¯�;

where is, as above, the Thurston measure on MF and where Œ� denotes the imageof in P MF . By definition, for X;Y 2 Tg , � 2MF , we have

dX

dY.Œ��/D

�pExt�.Y /pExt�.X/

�6g�6:

Note that the right-hand side only depends on Œ��.� Given that Œ�� 2P MF , let ˇ� W Tg � Tg !R be the cocycle defined by

ˇ�.X;Y /D log�pExt�.X/p

Ext�.Y /

�:

In our setting, ˇ� plays the role of the Busemann cocycle for the Teichmüllermetric, and (formally) the family ¹XºX2Tg of finite measures on P MF is afamily of conformal densities of dimension ıD 6g � 6 for the cocycle ˇ; thatis, for any X;Y 2 Tg and almost every � 2P MF , we have

dX

dY.Œ��/D exp

�ıˇ�.Y;X/

�and � �X .� �U /D X .U /;

where U �P MF and where � is an element of the mapping class group.Moreover, by the measure classification result of [LM], ¹XºX2Tg is theunique conformal density for the action of the mapping class group on Tgup to scale; the only possibility for ı is 6g � 6:

� Given that X 2 Tg and that transverse minimal foliations �; 2MF , define

ˇ.X; Œ��; Œ�/D�pExt�.X/Ext�.X/

A.�; /

�6g�6;

where A.�; / is the area of the quadratic differential with horizontal foliation� and vertical foliation . Note that the right-hand side only depends on Œ��, Œ�.Now introduce a measure P on the product P MF �P MF by the formula

PD ˇ.X; Œ��; Œ�/X.Œ��/X .Œ�/: (2.2)


Observe that the right-hand side does not depend on X and that it is invariantunder the diagonal action of the mapping class group. The space Q1Tg ofquadratic differentials fibers over P MF � P MF (with fiber R), and themeasure P on P MF �P MF lifts to (a multiple of) the Lebesgue measureon Q1Tg (by construction, it is invariant and belongs to the Lebesgue measureclass—but this determines the measure uniquely).

A consequence of Theorem 2.9 is that the measures X are the Patterson–Sullivanmeasures on Tg . These constructions are similar to those due to Kaimanovich in [Ka1]and [Ka2].

2.4. Measure in polar coordinatesRecall that Q1Tg is the space of pairs .X; q/, where X is a Riemann surface andwhere q is a holomorphic quadratic differential on X . (We occasionally drop X fromthe notation and denote points of Q1Tg by q alone.)

Fix X 2 Tg . Then we have a diffeomorphism ˆ W S.X/ � RC! Q1Tg , where

ˆ.q; t/ D gtq is the point in Q1Tg which is the endpoint of the length t geodesicsegment starting at X and tangent to q 2 S.X/. We can then write our measure dmin polar coordinates,

dm��.gtq/

�D .q; t/ dsX .q/dt: (2.3)

RemarkTo be completely precise, we should write dm.�.gtq// D ˆ�. .q; t/ dsX .q/dt/,but it is standard practice to avoid this formality with polar coordinates.

PROPOSITION 2.5Let K1 K be compact subsets of P .1; : : : ; 1/, q 2Q1Tg , and suppose that q andgtq both belong to �K . Then

j .q; t/j Ceht ; (2.4)

where C depends only on K . If in additionˇ̌®s 2 Œ0; t � W gsq 2 �K1

¯ˇ̌� .1=2/t; (2.5)

then there exists ˛ > 0 so that

.q; t/D eht��.q/�C.gtq/CO.e.h�˛/t /; (2.6)

where ˛ and the implied constant on (2.6) depend only on K1.

We prove this proposition in Section 3.2.


2.5. MixingLet U be an open subset of the boundary at infinity of Tg (which can be identi-fied with the space P MF of projective measured foliations). For each X 2 Tg , wemay identify U with a subset U.X/ of S.X/. Let SectU.X/ � Tg denote the setSt�0�.gtU.X//, that is, the sector based at X in the direction U. For a subset

A� Tg , Nbhdr.A/ denotes the set of points within the Teichmüller distance r of A.

THEOREM 2.6For any X 2 Tg and any � > 0, there exists an open subset U of S.X/ containing theintersection of S.X/ with the multiple zero locus such that for any compact K � Tg ,

lim supR!1

e�hRm�Nbhd1.BR.X/\ SectU.X/\ �K/

� �: (2.7)

Let K be a compact subset of Tg , and letK 0 be a compact subset of the principalstratum P .1; : : : ; 1/ � Q1Tg . Let BR.X;K;K 0/ denote the set of points Y 2 �K

such that d.X;Y / < R and the geodesic from X to Y spends at least half the timeoutside �K 0.

THEOREM 2.7For � > 0 and any compact K � Tg , there exists compact K 0 �P .1; : : : ; 1/�Q1Tgsuch that, for any X 2K ,

lim supR!1

e�hRm�Nbhd1.BR.X;K;K 0//

� �: (2.8)

Theorems 2.6 and 2.7 will be proved in Section 4.Let be a nonnegative compactly supported continuous function on Mg , which

we extend to a function WQ1Mg!R by making it constant on each sphere S.X/.We can consider to be a �-periodic function on Tg (or Q1Tg ). Let � be anothersuch function.

PROPOSITION 2.8 (mean ergodic theorem)We have

limR!1

he�hRZMg

�.X/�ZBR.X/

.Y /dm.Y /�dm.X/

D1

m.Mg/

ZMg

�.X/ƒ.X/dm.X/ZMg

.Y /ƒ.Y /dm.Y /: (2.9)


ProofSuppose that � > 0 is given. Let K be the union of the supports of � and . Withoutloss of generality, we may assume that the support of � is small enough so that thereexists an open U � @Tg such that, for each X in the support of �, U.X/ containsthe intersection of S.X/ with the multiple zero locus, and also such that (2.7) holds.Here, as above, U.X/ is the sector in S.X/ corresponding to directions in U.

We break up the integral over BR.X/ in (2.9) into two parts: the first overBR.X/\SectU.X/\�K and the second over the complement. The first is boundedby C1� in view of Theorem 2.7.

Let K 0 be as in Theorem 2.7. We use polar coordinates for the integral overBR.X/. We getZ

Mg

�.X/�ZBR.X/

.Y /dm.Y /�dm.X/

D

ZMg

�.X/�Z R

0

ZS.X/

.q; t/ .gtq/dsX .q/dt�dm.X/: (2.10)

In view of (2.8), we may assume (up to error of size C�ehR) that q 2 S.X/belongs to the compact set �K 00 D

SX2K S.X/ nU.X/, which is away from the

multiple zero locus. Also note that the left-hand side of (2.10) is symmetric in X andY (up to interchanging the functions � and ). So we may also assume that gtq alsobelongs to �K 00. Also in view of (2.8), we may assume that (2.5) holds. (Again, thecontribution of the part of the integral where this assumption is violated is boundedby C�ehR.)

We now consider the part of the region where none of the three assumptions areviolated; that is, q 2 �K 00, gtq 2 �K 00, and (2.5) holds. We are finally in a positionto use Proposition 2.5 to replace .q; t/ by eht��.q/�C.gtq/. Let c�˙ denote thetruncations of �˙ to �K 00. We can writeZ R

0

ehtZMg

�.X/

ZS.X/

c��.q/ .gtq/c�C.gtq/dsX .q/dm.X/dt

D

Z R

0

ehtZQ1Mg

�.q/c��.q/ .gtq/c�C.gtq/d�.q/dt; (2.11)

where we used the fact that � is constant on each sphere S.X/ and the formula

d�.q/D dm��.q/

�ds�.q/.q/:

Recall that �.Q1Mg/ <1 by Theorem 2.1. Now we can apply the mixing property

of the geodesic flow to the functions �c�� and c�C. The right-hand side of (2.11) isthen asymptotic to


ehR

h�.Q1Mg/

ZQ1Mg

�.q/c��.q/d�.q/ZQ1Mg

.q/c�C.q/d�.q/D

ehR

hm.Mg/

ZMg

�.X/ƒ.X/dm.X/ZMg

.Y /ƒ.Y /dm.Y /;

where the equality holds up to the truncation error. So the right-hand side of (2.9) isequal to the left-hand side of (2.9) up to an error which is bounded by C�, where Cdepends only on � and . Since � > 0 is arbitrary, the theorem follows. �

The proof of Proposition 2.8 also shows that, for any open U� S.X/,

limR!1

he�hRZMg

�.X/�ZBR.X/\SectU.X/

.Y /dm.Y /�dm.X/

D1

m.Mg/

ZMg

�.X/�Z

U

��.q/dsX .q/�dm.X/

ZMg

.Y /ƒ.Y /dm.Y /:

(2.12)

2.6. Counting

Proof of Theorem 1.2Let FR.X;Y / denote the cardinality of the intersection of BR.X/ with � � Y . Let �and be nonnegative compactly supported continuous functions on Mg . Note thatZ

Mg

FR.X;Y / .Y /dm.Y /DZBR.X/

.Y /dm.Y /:

Hence, by Proposition 2.8,

limR!1

e�hRZMg

�.X/

ZMg

FR.X;Y / .Y /dm.Y /dm.X/

D1

hm.Mg/

ZMg

�.X/ƒ.X/dm.X/ZMg

.Y /ƒ.Y /dm.Y /I

that is, e�hRFR.X;Y / converges weakly to 1hm.Mg/

ƒ.X/ƒ.Y /. We want to showthat the convergence is pointwise and uniform on compact sets.

Let � > 0 be arbitrary. Let be supported on a ball of radius � around Y (inthe Teichmüller distance) and satisfy

RMg

dmD 1. Let � be supported on a ball of

radius � around X , withRMg

� dmD 1. By the triangle inequality,ZMg

�.X/

ZBR�3�.X/

.Y /dm.Y /dm.X/

FR.X;Y /

ZMg

�.X/

ZBRC3�.X/

.Y /dm.Y /dm.X/:


After multiplying both sides by he�hR and letting R!1, we get, after applyingProposition 2.8,

e�3h�1

m.Mg/

ZMg

�.X/ƒ.X/dm.X/ZMg

.Y /ƒ.Y /dm.Y /

lim infR!1

he�hRFR.X;Y /

lim supR!1

he�hRFR.X;Y /

e3h�1

m.Mg/

ZMg

�.X/ƒ.X/dm.X/ZMg

.Y /ƒ.Y /dm.Y /:

Since � > 0 is arbitrary and since ƒ is uniformly continuous on compact sets, we get

limR!1

he�hRFR.X;Y /D1

m.Mg/ƒ.X/ƒ.Y /;

as required. �

In fact, the same argument using (2.12) instead of Proposition 2.8 yields the fol-lowing.

THEOREM 2.9Suppose thatX 2 Tg , and suppose that U is an open subset of the boundary at infinityof Tg . Then for any Y 2 Tg , we have, as R!1,

jBR.X/\ SectU.X/\ � � Y j �1

hm.Mg/ehRƒ.Y /

ZU

��.q/dsX .q/:

We also obtain the following.

THEOREM 2.10Suppose that X 2 Tg , and suppose that U and V are open subsets of the boundary atinfinity of Tg . Then for any Y 2 Tg , we have, as R!1,ˇ̌®

� 2 � W � � Y 2BR.X/\ SectU.X/ and ��1 �X 2 SectV .Y /¯ˇ̌

�1

hm.Mg/ehR

ZU

��.q/dsX .q/

ZV

�C.q/dsY .q/:

The proof of Theorem 2.10 is very similar to that of Theorem 2.9 except that wedo not assume that the function WQ1Mg !R is the pullback under � of a functionfrom Mg!R. The details are left to the reader.


3. The Hodge norm

3.1. The Hodge norm for abelian differentials

3.1.1. Definition and basic propertiesLetX be a Riemann surface. By definition,X has a complex structure. Let HX denotethe set of holomorphic 1-forms onX . One can define the Hodge inner product on HX

by

h!;i D

ZX

! ^ N:

We have a natural map r W H 1.X;R/!HX which sends a cohomology class � 2H 1.X;R/ to the holomorphic 1-form r.�/ 2 HX such that the real part of r.�/(which is a harmonic 1-form) represents �. We can thus define the Hodge inner prod-uct on H 1.X;R/ by h�1; �2i D hr.�1/; r.�2/i. We have

h�1; �2i D

ZX

�1 ^ ��2;

where � denotes the Hodge star operator, and we choose harmonic representatives of�1 and ��2 to evaluate the integral. We denote the associated norm by k � k. This isthe Hodge norm (see [FK]).

Let ˛ be a homology class in H1.X;R/. We can define the cohomology class�c˛ 2H

1.X;R/ so that, for all ! 2H 1.X;R/,Z˛

! D

ZX

! ^ �c˛:

Then, ZX

�c˛ ^ �cˇ D I.˛;ˇ/;

where I.�; �/ denotes the algebraic intersection number. We have, for any ! 2

H 1.X;R/,

h!; c˛i D

ZX

! ^ �c˛ D

Z˛

!:

We note that �c˛ is a purely topological construction which depends only on ˛, butc˛ depends also on the complex structure of X .

3.1.2. The Hodge norm and the hyperbolic metricLet ˛ be a simple closed curve on a Riemann surface X . Let `˛.�/ denote the lengthof the geodesic representative of ˛ in the hyperbolic metric which is in the conformalclass of X .


Fix �� > 0 (the Margulis constant) so that any two curves of hyperbolic lengthless than �� must be disjoint.

THEOREM 3.1For any constantD > 1, there exists a constant c > 1 such that, for any simple closedcurve ˛ with `˛.�/ <D,

1

c`˛.�/

1=2 kc˛k< c`˛.�/1=2: (3.1)

Furthermore, if `˛.�/ < �0 and if ˇ is the shortest simple closed curve crossing ˛,then

1

c`˛.�/

�1=2 kcˇk< c`˛.�/�1=2:

ProofLet ˛1; : : : ; ˛n be the curves with hyperbolic length less than �0. For 1 k n,let ˇk be the shortest curve with i.˛k; ˇk/ D 1, where i.�; �/ denotes the geometricintersection number. It is enough to prove (3.1) for the ˛k and the ˇk (the estimatefor other moderate length curves follows from a compactness argument).

We can find a collar region around ˛k as follows: take two annuli ¹zk W 1 > jzkj>jtkj

1=2º and ¹wk W 1 > wk > jtkj1=2º and identify the inner boundaries via the mapwk D tk=zk . (This coordinate system is used, e.g., in [Fa, Chapter 3], and also in[Ma1, Section 2], [Fo, Section 4], [W, Section 3], and elsewhere.) The hyperbolicmetric � in the collar region is approximately jdzj=.jzjj log jzjj/. Then `˛k .�/ �1=j log tkj. By [Fa, Chapter 3], any holomorphic 1-form ! can be written in the collarregion as

�a0.zk C tk=zk; tk/C

a1.zk C tk=zk; tk/

zk

�dzk;

where a0 and a1 are holomorphic in both variables. (We assume here that the limitsurface on the boundary of Teichmüller space is fixed.) This implies that, as tk! 0,

! D� azkC h.zk/CO.tk=z

2k/�dzk;

where h is a holomorphic function which remains bounded as tk! 0 and where theimplied constant is bounded as tk! 0. (Note that when jzkj � jtkj1=2, jtk=z2kj 1.)Now from the condition

R˛k�cˇk D 1, we see that on the collar of ˛j ,

cˇk C i�cˇk D� ıkj

.2�/zjC hkj .zj /CO.tj =z

2j /�dzj ; (3.2)


where the hkj are holomorphic and bounded as tj ! 0. (We use the notation ıkj D 1if k D j and zero otherwise.) Also from the condition

Rˇk�c˛k D 1, we have

c˛k C i�c˛k D1

j log tj j

�ıkjzjC skj .zj /CO.tj =z

2k/�dzj ; (3.3)

where skj also remains holomorphic and is bounded as tj ! 0. Then,

kcˇkk2 D k�cˇkk

2 D

ZX

cˇk ^ �cˇk �

Z 2�

0

Z 1

tk

1

4�2r2r dr d� �

j log tkj

4�

and

kc˛kk2 D k�c˛kk

2 D

ZX

c˛k ^ �c˛k �

Z 2�

0

Z 1

tk

1

.log tk/2r2r dr d� �

2�

j log tkj:

�3.1.3. The Hodge norm and extremal lengthWe recall the following theorem.

THEOREM 3.2 ([A, Theorem 1], [B, (1.4)])For any Riemann surface X and any � 2H1.X;R/,

kck2 D sup

�inf�2Œ

`� .�/2

Area.�/;

where the supremum is over the metrics � which are in the conformal class of X andwhere the infimum is over all the multicurves � homologous to �.

Note that in the definition (2.1) of extremal length, the infimum is over all thesimple closed curves � homotopic to �. Recall that the extremal metric � is alwaysthe flat metric obtained from a quadratic differential q, and in this metric the entiresurface consists of a flat cylinder in which � is the core curve. Thus, if this quadraticdifferential q is in fact the square of an abelian differential, then it follows from The-orem 3.2 that

Ext.X/D kck2: (3.4)

3.1.4. The Hodge norm and the geodesic flowLet �1Tg denote the space of (unit area) holomorphic 1-forms on surfaces of genusg. Recall that gt , the Teichmüller geodesic flow, preserves �1Tg (where we map�1Tg into Q1Tg by squaring abelian differentials). Fix a point S in �1Tg ; then S isa pair .X;!/, where ! is a holomorphic 1-form on X . Let k � k0 denote the Hodge


norm on the surface X0 D X , and let k � kt denote the Hodge norm on the surfaceXt D �.gtS/.

The following fundamental result is due to Forni.

THEOREM 3.3 ([Fo, Section 2])For any � 2H 1.X;R/ and any t � 0,

k�kt etk�k0:

Moreover, there exists a constant C depending only on the genus such that, if h�;!i D0 and, for some compact subset K of Mg , the geodesic segment ŒS; gtS� spends atleast half the time in ��1.K/, then we have

k�kt Ce.1�˛/tk�k0;

where ˛ > 0 depends only on K .

3.2. Proof of Proposition 2.5In this section we prove Proposition 2.5. We recall the setting: given that X 2 Tg ,let gtq be the point in Q1Tg which is the endpoint of the length t geodesic segmentstarting at X and tangent to q 2 S.X/. Then .q; t/ is given by

dm��.gtq/

�D .q; t/ dsX .q/dt:

Let K1 K be compact subsets of P .1; : : : ; 1/, q 2 Q1Tg , and suppose that q andgtq both belong to �K . Proposition 2.5 asserts a general bound, inequality (2.4),

j .q; t/j Ceht ;

where C depends only on K . If we are given some more information about the tra-jectory ¹gsqº0�s�t , in particular that it spends a portion of its time in a compact set,we can say more. Precisely, if inequality (2.5),ˇ̌®

s 2 Œ0; t � W gsq 2 �K1¯ˇ̌� .1=2/t;

is satisfied, then there exists ˛ > 0 so that inequality (2.6),

.q; t/D eht��.q/�C.gtq/CO.e.h�˛/t /;

holds. Moreover, ˛ and the implied constant in (2.6) depend only on K1.

ProofWithout loss of generality, q 2K . Let X D �.q/, and let Y D �.gtq/. Write Y D �z,where � 2 � and z 2 K . Let A be the matrix of � acting on the odd part of thehomology of the orienting double cover QX of q (i.e., A is the Kontsevitch–Zorichcocycle). Let h�; �i denote the Hodge inner product, and let A� denote the adjoint of


A, where we view A as a map between the inner product spaces determined by theHodge inner product at X and Y , respectively. Let nD hD dimH 1

odd.QX;R/, and let

e1; : : : ; en be an orthonormal basis for H 1odd.QX;R/ with respect to h�; �i so that

A�Aei D �2i ei ;

where et D �1 � � � � � �n D e�t > 0. Then kAeik D �i . Note that e1 is the imaginarypart of the holomorphic 1-form Q! on QX for which Q!2 D q and that en is the real partof Q!. It follows that �1 D et , �n D e�t . Also, since A is symplectic, �2 D ��1n�1.

Note that since we are assuming that q 2 K , the pair . QX; Q!/ associated to qbelongs to a compact subset (depending only on K) of the space of abelian differen-tials. Then by Theorem 3.3,

�2 DO.et / and ��1n�1 DO.e

t /; (3.5)

and if (2.5) holds,

�2 DO.e.1�˛/t / and ��1n�1 DO.e

.1�˛/t /: (3.6)

Since A has determinant 1, the product of the �i is 1.We can identify the tangent space to QTg withH 1

odd.QX;R2/DH 1

odd.QX;R/˝R2.

Let eCi D ei ˝ .10 / and e�i D ei ˝ .

01 / so that the eCi and e�i together form a basis

for the tangent space to QTg at q. Then .dgt /�.eCi / D e

tAeCi and .dgt /�.e�i / De�tAe�i . (In the above we adopt the convention that A acts on the tensor productH 1

odd.QX;R/ ˝ R

2 by acting on the H 1odd.QX;R/ factor, and trivially on the second

factor.)Let u1 D e

C1 � e

�n . We can complete the set ¹u1º to a basis ¹u1; : : : ; un�1º for

the tangent space of Q1Tg at q, so that for 2 i n� 1, ui D eCi Cwi , where wi is

in the span of ¹e�2 ; : : : ; e�n�1º. For consistency, we let w1 D�e�n . Note that the kwik

are bounded since q belongs to a compact set K , and the leaves of F u are transverseto the spheres S.X/ on K .

We extend the Hodge inner product h�; �i to H 1odd.QX;R2/ by declaring that heCi ;

e�j i D 0 for all i; j . This also extends the Hodge norm. We now estimate the follow-ing:

k.dgt /�.u1 ^ � � � ^ un�1/k2

D˝u1 ^ � � � ^ un�1; Œ.dgt /��

�Œ.dgt /��u1 ^ � � � ^ un�1˛

DXP

Du1 ^ � � � ^ un�1; Œ.dgt /��

�Œ.dgt /��î…P

eCi ^î2P

wi

�E

DXP

Du1 ^ � � � ^ un�1;

�î…P

e2t�2i eCi ^

î2P

e�2tA�Awi

�E; (3.7)


where Œ.dgt /�� is the adjoint of the linear transformation .dgt /�, and the sum is overall subsets P of ¹1; : : : ; n� 1º.

Note that by (3.5), for 1 i n� 1,

ke�2tA�Awik C Cke2t�2i k:

Thus all the terms in (3.7) are bounded by Ce2t.n�1/�21Qn�1iD2 �

2i D Ce

2nt D Ce2ht .Thus,

k.dgt /�.u1 ^ � � � ^ un�1/k DO.eht /: (3.8)

By definition, .q; t/ is the determinant of the linear transformation D D

.d�Y /�.dgt /�. Since Y is in a compact set, the norm of .d�Y /� is bounded. Thistogether with (3.8) implies (2.4). If we assume (3.6), we get, for any 1 i n� 1,

ke�2tA�Awik Ce�2˛t Ce�2˛te2t�2i ;

which implies that the contribution of all the terms in (3.7) to .q; t/, except for theterm withP D;, isO.e.h�˛/t /. It remains to evaluate the term with P D;. Droppingthe rest of the terms is equivalent to replacing the map ˆ.q; t/ by the composition ofthree maps:(1) projecting from the sphere S.X/ to the leaf of the expanding foliation F u

passing through q;(2) flowing by time t ;(3) applying the map �Y to project from the leaf of F u through gtq to the Teich-

müller space Tg .The Jacobian of the resulting map is the product of the Jacobians from steps (1), (2),and (3). Step (1) gives a factor of ��.q/, step (2) gives a factor of eht , and step (3)gives a factor of �C.gtq/, so one gets formula (2.6). �

3.3. The Hodge norm and quadratic differentials

3.3.1. The Hodge norm and the flat metricTheorem 3.1 allows us to estimate the Hodge norm in terms of the hyperbolic metric.In this section we state a lemma which allows us to estimate the Hodge norm in termsof the flat metric. This is done by first estimating the hyperbolic metric in terms of theflat metric and then using Theorem 3.1.

Let q be a holomorphic quadratic differential, and let `q denote length in the flatmetric defined by q. For each � > 0, let K.�/ denote the complex generated by allsaddle connections shorter than � (see [EMa, Section 6]). Recall that K.�/ containsall saddle connections shorter than � and that `q.@K.�//DO.�/, where the impliedconstant depends only on the genus. (Also, if C is a cylinder whose boundary is


shorter than �, then we include C in K.�/.) Let �1 < � � �< �n denote the values of �,where K.�/ changes. (Note that n is bounded in terms of the genus.) Pick a constantC � 1 and drop all �i such that �i > �iC1=C . After renumbering, we obtain new �i

1 i m, with �i < �iC1=C . To simplify notation, we let �mC1 D 1. For 1 i m,let Ki denote the complex K.�i /.

LEMMA 3.4 (see Rafi [R2])Let q be a holomorphic quadratic differential on a surface †g , and let � be thehyperbolic metric on †g in the same conformal class as q. Choose C � 1, and letthe �i , 1 i m and the complexes Ki be as in the preceding paragraph. Then wehave the following.(a) There exists a constant �0 depending only on C and the genus (with �0! 0

as C !1) such that any simple closed curve ˛ on †g with the hyperboliclength `˛.�/ < �0 is homotopic to a connected component of the boundary ofone of the Ki .

(b) Let � be a connected component of @Ki . If � is not a core curve of a flatannulus, then `� .�/� C1=j log �i j, where C1 is a constant depending only onthe genus.

3.3.2. The modified Hodge norm and quadratic differentialsTheorem 3.3 gives a partial hyperbolicity property of the geodesic flow on spaces ofabelian differentials. In our applications, we need a similar property for the spacesQ1Tg of quadratic differentials. A standard construction, given X 2 Tg and q a holo-morphic quadratic differential on X , is to pass to the possibly ramified double coveron which the foliation defined by q is orientable. This yields a surface QX with a holo-morphic abelian differential !. However, a major difficulty is the following: even if qbelongs to a compact subset of Q1Tg , the complex structure on QX may have very shortclosed curves in the hyperbolic metric. This may occur since even if one restricts q tocompact subsets of Teichmüller space, the flat structure defined by q may have arbi-trarily short saddle connections (connecting distinct zeros). Such a saddle connectionmay lift to a very short loop when one takes a double cover. Other more complicatedtypes of degeneration are also possible. However, the following simple observation iskey to our approach.

LEMMA 3.5For every compact subset K �Mg , there exists a constant �0 > 0 depending onlyon K such that, for any q 2 ��1.K/, the flat structure associated to the holomor-phic quadratic differential on the orienting cover of q has no closed trajectories ofEuclidean length less than �0 which are part of a flat cylinder.


Together with Lemma 3.4(b), Lemma 3.5 will allow us to estimate the hyperboliclength of short curves on QX .

Proof of Lemma 3.5Given the set K , there exists a constant �0 > 0 such that any loop of length less than�0 in the flat metric is contractible. Now suppose that q 2 ��1.K/, and suppose that� is a trajectory on the associated orienting double cover which has length less than�0 and is part of a flat cylinder C . Let �1 be the projection of � to the original flatstructure defined by q. Then �1 must also be a part of a flat cylinder, and the lengthof �1 must be at most �0. Then from the definition of �0, it follows that �1 must becontractible. But this is impossible since the curvature of �1 is zero, and we are notallowing q to have poles. �

Short bases. Suppose that .X;!/ 2 �1Tg (where the notation �1Tg is definedas in Section 3.1.4). Fix �1 < �� (where �� is the Margulis constant defined in Sec-tion 3.1.2), and let ˛1; : : : ; ˛k be the curves with hyperbolic length less than �1 onX . For 1 i k, let ˇi be the shortest curve in the flat metric defined by ! withi.˛i ; ˇi /D 1. Let �r , 1 r 2g � 2k be moderate length curves on X so that the˛j , ˇj , and �j are a symplectic basis S forH1.X;R/. We will call such a basis short.

The functions �i .!/ and the modified Hodge norm. We would like to use theHodge norm on double covers of surfaces in Q1Tg . One difficulty is that if one takesthe double cover of some quadratic differential in the multiple zero locus, the Hodgenorm in some directions tangent to the multiple zero locus might vanish. As a con-sequence, if we use the Hodge norm to define a Hodge distance on Q1Tg , then wefind that some balls are not compact and that the resulting distance does not separatepoints on the multiple zero locus.

We now define a modification of the Hodge norm in order to avoid these problems.The modified norm is defined on the tangent space to the space of pairs .X;!/, whereX is a Riemann surface and ! is a holomorphic 1-form onX . Unlike the Hodge norm,the modified Hodge norm will depend not only on the complex structure onX but alsoon the choice of a holomorphic 1-form ! onX . Let ¹˛i ; ˇi ; �rº1�i�k;1�r�2g�2k be ashort basis for .X;!/. For 1 i k, we define �i .!/ to be zero if ˛i has a flatannulus in the flat metric defined by ! and 1 otherwise.

We can write any � 2H 1.X;R/ as

� D

kXiD1

ai .�c˛i /C

kXiD1

bi`˛i .�/1=2.�cˇi /C

2g�2kXrD1

ur.�c�r /: (3.9)

(Recall that � denotes the hyperbolic metric in the conformal class of X , and for acurve ˛ on X , that `˛.�/ denotes the length of ˛ in the metric � .) We then define


k�k0 D k�kC� kXiD1

�i .!/jai j C

kXiD1

jbi j C

2g�2kXrD1

jur j�: (3.10)

From (3.10), we have for 1 i k,

k�c˛i k0 � 1; (3.11)

as long as ˛i has no flat annulus in the metric defined by !. Similarly, from (3.2) wehave

k�cˇik0 � k�cˇik �

1

`˛i .�/1=2: (3.12)

In addition, in view of Theorem 3.1, if � is any other moderate length curve on X ,then k�c�k0 � k�c�k D O.1/. Thus, if B is a short basis associated to !, then forany � 2B,

Ext� .!/1=2 � k�c�k k�c�k

0 (3.13)

(see (3.4)). (By Ext� .!/ we mean the extremal length of � in X , the conformal struc-ture defined by !.)

RemarkFrom the construction, we see that the modified Hodge norm is greater than the Hodgenorm. Also, if the flat length of the shortest curve in the flat metric defined by ! isgreater than �1, then for any cohomology class �, for some C depending on �1 andthe genus, we have

k�k0 Ck�kI (3.14)

that is, the modified Hodge norm is within a multiplicative constant of the Hodgenorm.

3.4. Nonexpansion of the modified Hodge normIn this section we prove a weak version of Theorem 3.3 for the modified Hodge norm.

Let k � k00 denote the modified Hodge norm on the surface S0 D S D .X0;!0/,and let k � k0t denote the modified Hodge norm on the surface St D gtS D .Xt ;!t /.

THEOREM 3.6There exists a constant C depending only on the genus and on �1 such that, for any� 2H 1.X;R/ and any t � 0,

k�k0t Cetk�k00: (3.15)

The proof of Theorem 3.6 is based on the following lemma.


LEMMA 3.7 (see Rafi [R1], [R2])Suppose that .X0;!0/ 2�1Tg , and write .Xt ;!t / for gt .X0;!0/. Suppose that B isa short basis in the flat structure defined by !0 and that B 0 is a short basis in the flatstructure defined by !t . Suppose that ˛ is a curve of hyperbolic length < �1 on bothX0 and Xt (so that in particular, ˛ 2B and ˛ 2B 0), and that ˛ has no flat annulus.Let ˇ0 2B 0 denote the curve with i.˛;ˇ0/D 1. Then, for any � 2B, we have

i.�;ˇ0/ CetExt� .X0/1=2;

where i.�; �/ denotes the intersection number.

Recall that, given simple closed curves ˛ and ˇ on †g , the intersection numberi.˛;ˇ/ is the minimum number of points in which representatives of ˛ and ˇ mustintersect.

Proof of Lemma 3.7First, we claim that i.�;ˇ0/ i.�;P /, where P is the shortest pants decompositionof Xt . This is because either � is disjoint from ˛, or if it intersects ˛ then the rel-ative twisting of � and ˇ0 is bounded (see [R1, Section 4]). But this shortest pantsdecomposition has extremal length less than a constant times e2t in X0. Using [Mi1,Lemma 5.1], we have

i.�;P /2 Ext� .X0/ExtP .X0/ C2e2tExt� .X0/:

This finishes the proof. �

Proof of Theorem 3.6Let ˛01; : : : ; ˛

0k

be the curves with hyperbolic length less than �1 on Xt D �.St /. Letˇ01; : : : ; ˇ

0k

be the shortest curves in the Euclidean metric on St such that i.˛r ; ˇj /Dırj . Let � 0r , 1 r 2g � k be moderate length curves on Xt so that the ˛j , ˇ0j ,and � 0j are a symplectic basis S 0 for H1.X;R/. Thus S 0 is a short basis. For any� 2H 1.X;R/, we may write

� D

kXiD1

ai .�c˛0i/C

kXiD1

bi`˛0i.�/1=2.�cˇ 0

i/C

2g�kXrD1

ui .�c� 0r /: (3.16)

In view of (3.10) and Theorem 3.3, it is enough to show that

kXiD1

�i .!t /jai j C

kXiD1

jbi j C

2g�kXrD1

jur j Cetk�k00: (3.17)

We have


bi D1

`˛0i.�/1=2

hc˛0i; �it

1

`˛0i.�/1=2

kc˛0iktk�kt k�kt Ce

tk�k0 Cetk�k00:

Here, h�; �it denotes the Hodge inner product on Xt . Also, let ı0i 2 S 0 be the elementwith i.� 0i ; ı

0i /D 1. Then,

jui j D hcı0i; �it kcı0

iktk�kt C1k�kt C2e

tk�k0 C3etk�k00;

where C1, C2, and C3 depend only on the genus. It remains to bound jai j. We mayassume that ˛i has length less than �1 on X0 as well. (If ˛i is longer than �1 on X0,we can put in an intermediate point X 00, where ˛i has length � �1.)

Let S D ¹˛i ; ˇi ; �iº be a short basis for S0. It is enough to prove (3.17) assumingthat � D�c for some � 2 S . Then,

ai D h�cˇ 0i; ci D I.�;ˇ

0i /:

Since the algebraic intersection number I.�; �/ is bounded by the geometric intersec-tion number i.�; �/, we have

jai j D jI.�;ˇ0i /j i.�;ˇ

0i / Ce

tExt.X0/1=2 Cetk�ck

0;

where we have used Lemma 3.7 and (3.13). �

3.5. The Euclidean, Teichmüller, and Hodge distancesNote that we can locally identify a leaf of F s (or F u) with a subspace of H 1. QX;R/,where QX is the double cover of X 2 Tg . If � is a map from Œ0; r� into some leaf ofF ss , then we define the (modified) Hodge length `.�/ of � as

R r0k� 0.t/k0 dt , where

k � k0 is the modified Hodge norm. If q and q0 belong to the same leaf of F ss , thenwe define the Hodge distance dH .q; q0/ to be the infimum of `.�/, where � variesover paths connecting q and q0 and staying in the leaf of F ss containing q and q0. Wemake the same definition if q and q0 are on the same leaf of F uu.

In what follows, K is a compact subset of Tg , and all implied constants dependon K . Given the set K , there exists a constant �0 > 0 such that, for any q 2 ��1.K/,any loop of length less than �0 in the flat metric given by q is contractible.

For each q 2 ��1.K/, there exists a canonical (marked) Delaunay triangulationof the flat metric given by q (see [MaS, Section 4]). Since there are only finitely manycombinatorial types of (unmarked) triangulations on a genus g surface and since themapping class group � acts properly discontinuously, there are only finitely manycombinatorial types ˛ of marked Delaunay triangulations on ��1.K/. Call this set I .We have

��1.K/DG˛2I

W˛;


where W˛ is the subset of ��1.K/ and where the combinatorial type of the markedDelaunay triangulation is given by ˛. Let J � I denote the subset such that, for˛ 2 J , W˛ is relatively open in ��1.K/. Then we may write

��1.K/DG˛2J

W 0˛;

whereW˛ W 0˛ W˛ . If we choose a basis S forH odd1 . QX;Z/, then we have a period

map ˆS W Q1Tg ! C

h given by integrating the canonical square root of q alongthe chosen basis. Recall that this is a local coordinate system for Q1Tg . In fact, thefollowing holds.

LEMMA 3.8For each ˛ 2 J , there exists a basis S˛ of H odd

1 . QX;Z/ consisting of lifts of edges ofthe Delaunay triangulation such that the map ˆ˛ DˆS˛ is defined on all of W 0˛ andis a coordinate system on the part of W 0˛ not contained in the multiple zero locus.

ProofSee [MaS, Section 4]. �

We can now define a Euclidean norm on the tangent space of Q1Tg as follows.If v is a tangent vector based at the point q 2W˛ , then we define

kvkE D jˆ�˛.v/j;

where j � j is the standard norm on Ch. The norm k � kE depends on the choices of

the S˛ and the W 0˛ ; however, if k � k0E is the Euclidean norm obtained from differentchoices, then the ratio of k � kE and k � k0E is bounded by a constant depending onlyon K .

The Euclidean distance. We define the Euclidean length of a path � W Œ0; r�!Q1Tg to be

R r0k� 0.t/kE dt . We define the Euclidean distance dE .q; q0/ to be the

infimum of the Euclidean length of the paths connecting q and q0. Locally, up to amultiplicative constant, dE .q; q0/D jˆ˛.q/�ˆ˛.q0/j.

We denote the Teichmüller distance on Tg by dT . For p 2 Q1Tg and � > 0,let BE .p; �/ denote the ball in the Euclidean metric of radius � centered at p, andlet BT .�.p/; �/ denote the analogous ball in the Teichmüller metric. We note thefollowing.

LEMMA 3.9There exist continuous functions fi W RC! R

C with fi .r/! 0 as r ! 0 such that,for any two points p1 and p2 in ��1.K/ on the same leaf of F uu, we have


f1�dE .p1; p2/

� dT

��.p1/;�.p2/

� f2

�dE .p1; p2/

�:

ProofThis follows from the compactness of K . �

RemarkThe Euclidean distance has a number of useful properties: it behaves well near themultiple zero locus, and on compact subsets of Tg it is absolutely continuous withrespect to the Teichmüller distance (see Lemma 3.9). However, often we need to dealinstead with the modified Hodge distance because of its nonexpansion and decayproperties under the geodesic flow (see Theorem 3.15 below).

The following theorem is the main result of this section.

THEOREM 3.10There exist constants 0 < c1 < c2, both depending only on K , such that for anyq1; q2 2 �

�1.K/ on the same leaf of F uu with dE .q1; q2/ < 1, we have

c1dE .q1; q2/ dH .q1; q2/ c2dE .q1; q2/j logdE .q1; q2/j1=2:

Thus, in particular, on compact sets, the modified Hodge metric is equivalent tothe Euclidean metric, and then, in view of Lemma 3.9, the modified Hodge metric onthe restriction of a leaf of F uu to ��1.K/ is equivalent to the Teichmüller metric.

Note that the lower bound in Theorem 3.10 is clear since, up to a constant depend-ing only on K , the modified Hodge norm is always bigger than the Euclidean norm(see (3.11), (3.12)). The rest of this section consists of the proof of the upper boundin Theorem 3.10. For q 2Q1Tg , let `min.q/ denote the length of the shortest saddleconnection in the flat metric defined by q.

We use the following standard fact about Delaunay triangulations.

LEMMA 3.11Suppose that T is the Delaunay triangulation of a surface in q 2Q1Tg . Let `min.q/

denote the length of the shortest saddle connection in q. Then any saddle connectionin q of length at most

p2`min.q/ is an edge of T .

ProofLet z1; : : : ; zn denote the zeros of q (i.e., the conical points in the flat metric). Foreach zi , let the Voronoi polygon Vi denote the set of points of q that are closer to zithan to any other zj , j ¤ i .


Figure 1. Proof of Lemma 3.11

Recall that the Delaunay triangulation T of q is dual to the Voronoi diagram inthe sense that zi and zj are connected by an edge of T if Vi and Vj share a commonedge (and also under some conditions if Vi and Vj share a common vertex).

Let e be a saddle connection in q connecting zi and zj (see Figure 1). Let mbe the midpoint of e. Let k be such that d.zk;m/ is minimal. Suppose that e is notan edge of the Delaunay triangulation T . Then k ¤ i; j and d.m; zk/ d.m; zi /D.1=2/d.zi ; zj /. We may assume that the angle at m between the segments mzi andmzk is smaller than �=2 (otherwise replace zi by zj ). Let p be the point on thesegmentmzi such that d.p;m/D d.p; zk/. Now consider the isosceles acute triangle� whose vertices are m, p, and zk . This triangle cannot contain any zeros (or else zkwould not be the saddle connection closest tom). Therefore, d.p; zk/

p2d.m; zk/.

Hence,

`min.q/ d.zi ; zk/ d.zi ; p/C d.p; zk/�d.zi ;m/� d.m; zk/

�Cp2d.m; zk/

D1

2d.zi ; zj /C .

p2� 1/d.m; zk/

1

2d.zi ; zj /C

.p2� 1/

2d.zi ; zj /

D

p2

2d.zi ; zj /:

Thus, if e is not an edge of the Delaunay triangulation, then `.e/�p2`min.q/. �

Recall that an integral multicurve is a finite set of oriented simple closed curveswith integral weights. By convention, a negative weight corresponds to reversing theorientation.


LEMMA 3.12Suppose that T is a geodesic triangulation of an orientable surface ! in �1Tg . Weorient each edge of T so that its x component is positive (for a vertical edge theorientation is not defined). Suppose that W is a subset of the edges of T . Then thereexists an integral multicurve on ! such that(a) is disjoint from the vertices of T and is transverse to the edges;(b) crosses each edge of W at least once;(c) each time crosses an edge of e 2W , the crossing is from left to right (with

respect to the orientation on the edge); if e is vertical, then all crossings mustbe from the same side;

(d) for each edge e of T , the intersection number i. ; e/ n, where n dependsonly on the genus g.

ProofEach triangulated surface which has vertical edges is the limit of triangulated surfaceswhich do not. Hence, without loss of generality, we may assume that T has no verticaledges.

We define a directed graph G as follows (see Figure 2). The vertices of G arethe connected components of ! nW . Two vertices A and B of G are connected by adirected edge of G if and only if there exists a saddle connection � 2W such that Ais incident to � from the left and B is incident to � from the right.

We now claim that, for each pair A, B of vertices of G , there exists at least onedirected path from A to B . This can be derived from the minimality of the flow in

Figure 2. Lemma 3.12. The surface consists of three flat tori glued to each other as shown. The setW consists of the three dotted lines separating the tori. (The two dotted lines on the opposite sidesof the figure are identified.) The multicurve (in this case closed curve) is drawn. The black dotsare vertices of the graph G (used in the proof).


an almost vertical direction, but we prefer to give a direct combinatorial argument.Suppose that this is not true. Let A be the set of vertices of G which can be reachedfrom A by a directed path. Then A 2 A and B 2 Ac ; in particular, A and Ac areboth nonempty. Now let D be the closure of the connected components in ! n Wcorresponding to vertices of A. Then D is a subsurface of !, and @D consists ofedges from W . For each edge � 2 @D, let �� DC1 if D is on the left of � and �1otherwise. Then (since D is orientable),

P�2@D �� D 0. It follows that there exists

� 2 @D such that the x component of �� is positive. Since the x component of �is positive, this implies that �� D 1. Thus the directed edge of G corresponding to �is directed from D to Dc . This means that some vertex of G outside of A can bereached from a vertex in A by a directed path, contradicting the definition of A.

Thus, every pair of vertices of G can be connected by a directed path. This impliesthat every directed edge of G is contained in a directed cycle of G . Therefore, thereexists a finite union of directed cycles such that every edge in G is contained in .The length of is bounded in terms of the size of G , that is, in terms of the genus.We can thus realize as a multicurve on ! with properties (a)–(d). �

LEMMA 3.13There exist �1 > 0 and C > 0 depending only on K such that, for any q 2 ��1.K/,there exist q0 2 ��1.K/\F uu.q/ and a path � W Œ0; �1�!F uu.q/ such that �.0/Dq, �.�1/D q0, and for t 2 Œ0; �1�,

`min��.t/

�� `min

��.0/

�p1C t2; (3.18)

and also

dH .q; q0/ C

Z �1

0

ˇ̌log�`min.q/

p1C t2

�ˇ̌1=2dt: (3.19)

ProofLet QS D . QX; Q!/ denote the double cover corresponding to q. Consider the Delau-nay triangulation of QS . By Lemma 3.11, all saddle connections of length at mostp2`min. QS/ belong to the Delaunay triangulation. Let W denote this set of saddle

connections. Let be the multicurve obtained by applying Lemma 3.12 to QS and W .Let � be the involution corresponding to q. Let 0 D � �. /. Then 0 also has

properties (a)–(d) of Lemma 3.12. In addition, �. 0/D� 0.Let �1 2 .0;

p2/ be a constant which will be chosen later in this proof (depending

only on the genus). We now use 0 to define a path � W Œ0; �1�!Q1Tg contained in��1.K/ and staying on the same leaf of F uu. We will actually define the path Q�.t/,where for each t , Q�.t/ is the double cover of �.t/. Let QS denote the double cover of q,and set Q�.0/D QS . For each t 2 Œ0; �1�, Q�.t/ is built from the same triangles as QS , but


for each edge e, we add I.e; 0/t`min.q/ to the x-component, where I.�; �/ denotesthe algebraic intersection number (see [MaS, Section 6]).

Note that we do not assert that the Delaunay triangulation of Q�.t/ is the same asthat of QS D Q�.0/. However, because of the form of 0, the involution � acts on Q�.t/,and we can let �.t/ be the quotient by � .

We now claim that (3.18) holds. Indeed, if e 2W is a saddle connection in Q�.0/with vector .x0; y0/, then x0 � 0, and on Q�.t/, e has lengthq

y20 C�x0C I.e; 0/`min.q/t

�2�p`min.q/2C `min.q/2t2

D `min.q/p1C t2: (3.20)

Suppose that t1; t2 2 Œ0; �1� and are such that is a saddle connection on Q�.t/ fort1 < t < t2. Let `t ./ denote the length of in the flat metric on Q�.t/. Then,

`t ./� `t1./� jI.; /j`min.q/.t � t1/: (3.21)

Now suppose that z andw are any two zeros of q. Let �t .z;w/ denote the shortestpath between z and w on Q�.t/, and let j�t .z;w/j denote the length of �t .z;w/, that is,the distance between z and w in the flat metric on Q�.t/. Suppose that �0.z;w/ is notan edge in W . Then either �0.z;w/ is a saddle connection not in W , in which casej�0.z;w/j �

p2`min.q/, or �0.z;w/ is a union of at least two saddle connections,

so that j�0.z;w/j � 2`min.q/ �p2`min.q/. It now follows from (3.21) that for all

t 2 Œ0; �1�,

j�t .z;w/j � .p2� nmt/`min.q/;

where n is the maximum intersection number of 0 with a saddle connection in theDelaunay triangulation of QS and where m is the maximal number of saddle connec-tions in �t .z;w/. Note that both n and m are bounded by the genus. Now we choose�1 so that

p2 � nm�1 � .1C �1/

2. Then, if �0.z;w/ is not a saddle connection inW , then for all t 2 Œ0; �1�,

j�t .z;w/j � `min��.0/

�p1C t2: (3.22)

Now (3.18) follows from (3.20) and (3.22).We now estimate the Hodge length of the path � W Œ0; �1�! Q1Tg . By Lem-

mas 3.4 and 3.5,

`�.t/.�/�Cg

j log.`min.q/p1C t2/j

;

where `� .S/ is defined to be the infimum over all simple closed curves ˛ of `˛.�/(and � is the hyperbolic metric in the conformal class of S ). By construction, the


intersection number of 0 with a short basis (see Section 3.3.2) of �.t/ is boundeddepending only on the genus. Therefore, by (3.9) and (3.10),

k� 0.t/k0 Cˇ̌log�`min.q/

p1C t2

�ˇ̌1=2;

where C depends only on the genus. Thus, if q1 D �.�1/, then

dH .q; q1/

Z �1

0

k� 0.t/k0 dt

Z �1

0

Cˇ̌log�`min.q/

p1C t2

�ˇ̌1=2dt:

Thus (3.19) holds. �

LEMMA 3.14There exists �0 > 0 (depending on K) such that, for all � < �0 and for all q 2 ��1.K/

with `min.q/ < �, there exists q0 on the same leaf of F uu as q with `min.q0/� �, and

dH .q; q0/ k�j log �j1=2, where k depends only on K .

ProofWe define a sequence qn as follows. Let q0 D q. If qn has been defined already, weapply Lemma 3.13 with q D qn, and we define qnC1 to be the point q0 guaranteed byLemma 3.13. We obtain a sequence qn with

`min.qnC1/� `min.qn/.1C �21/1=2 D �2`min.qn/;

where we let �2 D .1 C �21/1=2. Define tn inductively by t0 D 0, tnC1 D tn C

�1`min.qn/, n� 0. Then,

tn D �1

n�1XkD0

`min.qk/ �1

n�1XkD0

`min.qn/

�n�k2

D �1`min.qn/��12 � �

�n�12

1� ��12�1`min.qn/

�2 � 1:

Then,

tnC1 D tnC �1`min.qn/�1�2

�2 � 1`min.qn/:

Thus, for t 2 Œtn; tnC1�,

`min.qn/��2 � 1

�1�2t � �3t:

We have

dH .qn; qnC1/

Z tnC1

tn

Cˇ̌log�`min.qn/

p1C t2

�ˇ̌1=2dt

Z tnC1

tn

C j log.�3tp1C t2/j1=2 dt:


Thus,

dH .q0; qn/

Z tn

0

C j log.�3tp1C t2/j1=2 dt:

We choose n so that `min.qn/ is comparable to �. Let q0 D �.tn/. Then, `min.q0/ �

tn � �. Now the modified Hodge length of the path q0; q1; : : : ; qn is O.�j log �j1=2/.�

Proof of Theorem 3.10Since K is compact, the intersection of the multiple zero locus with ��1.K/ is (con-tained in) a finite union of hyperplanesH1; : : : ;Hn. Each hyperplaneHj has complexcodimension 1. We can choose �0 > 0 such that any two Hj which do not intersect in��1.K/ are at least �0 apart. Let ı0 > 0 be a lower bound on the angle between anytwo Hj which do intersect in ��1.K/. Clearly, �0 and ı0 depend only on K .

Let Z be the locus where our quadratic differential has either a zero of order atleast 3 or two zeros each of order at least 2. Then there exists a constant k0 (dependingonly on ı0 and thus only on K) such that for any q 2 ��1.K/ and any � > 0, if theball BE .q; �/ intersects at least two hyperplanes Hj , then dE .q;Z/ < k0�.

Take two points q1; q2 2 ��1.K/ on the same leaf of F uu, with d.q1; q2/D �.Choose k3 > k0 (thus k3 depends only on K). We also assume that k3� < �0. ByLemma 3.14 (with k3� in place of �), there exist q01; q

02 so that for i D 1; 2, we

have dH .qi ; q0i / k1�j log �j1=2, dE .qi ; q0i / k2�, and `min.qi /D k3�, where k1; k2depend only on K . Since k3 > k0 and k3� < �0, there exist �1, �01 such that k3�=2 >�1 > �

01 > �=.2k3/, and for all q 2 ��1.K/, either dE .q;Z/ < �1 or BE .q; �01/ con-

tains at most one hyperplane from the multiple zero locus.Note that the intersection ofZ with ��1.K/ has complex codimension at least 2.

Hence, the intersection of the �1-neighborhood ofZ with F uu.q1/\��1.K/ is con-

tained in the �1-neighborhood of a finite union of hyperplanes, each of real codimen-sion at least 2. Then there exists a constant k4 depending only on K , and a path �connecting q01 to q02 of length at most k4�, which avoids the �1-neighborhood of Z.

Now let p0 D q01, and mark points pi along � which are �01=2 apart in theEuclidean metric. We have pn D q02. Let Bi be the ball of Euclidean diameter �01=2which contains pi and piC1 on its boundary. By construction, Bi contains at mostone hyperplane (which we will denote L) from the multiple zero locus. Note that byLemma 3.4, Lemma 3.5, and (3.12), for any p 2Bi and any tangent vector v at p, themodified Hodge norm of v can be estimated as

kvk0H C j logdE .p;L/j1=2kvkE ; (3.23)

where kvkE is the Euclidean norm of v and where d.p;L/ denotes the Euclideandistance between the point p and the hyperplane L.


Let p0i be the farthest point in Bi from the hyperplane. Then, after connect-ing pi and p0i by a straight line path and using (3.23), we see that dH .pi ; p0i / DO.�j log �j1=2/ and also that dH .p0i ; piC1/DO.�j log �j1=2/. Thus, since the numberof Bi along the path is bounded by a constant depending only on K , we finally obtain

dH .q01; q02/DO

�dE .q

01; q02/j log �j1=2

�: �

3.6. The nonexpansion and decay of the Hodge distance

THEOREM 3.15Suppose that q 2Q1Tg and q0 2Q1Tg are in the same leaf of F ss . Then we have thefollowing.(a) There exists a constant cH > 0 such that, for all t � 0,

dH .gtq;gtq0/ cHdH .q; q

0/:

(b) Suppose that � > 2�1, where �1 is as in the definition of short basis (see Section3.3.2). Let K D ¹q 2 Q1Tg W `min.q/ > �º. Suppose that dH .q; q0/ < 1, andsuppose that t > 0 is such thatˇ̌®

s 2 Œ0; t � W gsq 2 �K¯ˇ̌� .1=2/t: (3.24)

Then for all 0 < s < t ,

dH .gsq;gsq0/ Ce�csdH .q; q

0/;

where c and C depend only on g and �.

ProofStatement (a) follows immediately from Theorem 3.6. For the second statement, let� W Œ0; ��! Q1Tg be a modified Hodge length minimizing path connecting q to q0

(and staying in the same leaf of F ss). We assume that � is parameterized so thatdH .�.u/; q/D u. By assumption, � < 1.

Let N D 4cH=.c1��/, where c1 is as in Theorem 3.10. For 0 j N , let uj Djc1�=.4cH /. We now claim that for all j there exists tj such that for s > tj such thatgsq 2 �K , we have, for all 0 u < uj ,

dE�gs�.u/; gsq

�< �

jXkD0

2�k (3.25)

and

dH�gs�.u/; gsq

�<Ce�csu; (3.26)


where C and c depend only on g and �. The equations (3.25) and (3.26) will beproved by induction on j . If j D 0, then there is nothing to prove. Now assume thatj � 1 and that (3.25) and (3.26) are true for all u uj�1. Suppose that u < uj . By(a), we have, for all t > 0,

dH�gt�.u/; gt�.uj�1/

�< cH .u� uj�1/ < c1�=4:

Therefore, by Theorem 3.10, dE .gt�.u/; gt�.uj�1// < �=2. Therefore, by the induc-tive assumption (3.25), for the s > tj�1 such that gsq 2 �K , for all uj�1 < u < uj ,we have `min.gs�.u// > �=2. Thus, by (3.14), for all uj�1 < u < uj , the modifiedHodge norm of � 0.u/ is within a constant of the Hodge norm of � 0.u/. Then by The-orem 3.3, for all uj�1 < u< uj ,

dH�gt�.u/; gt�.uj�1/

�<Ce�ctdH

��.u/; �.uj�1/

�: (3.27)

Therefore, by Theorem 3.10, there exists tj > 0 (depending only on g and �) suchthat, for s > tj with gsq 2 �K , and for uj�1 u < uj , (3.25) holds. Also, (3.26)follows from (3.27). The induction terminates after finitely many steps (dependingonly on g and �). Thus (b) holds. �

4. The multiple zero locusIn this section we prove Theorems 2.6 and 2.7.

Let K be a compact subset of Tg . We assume that K is contained in one funda-mental domain for the action of � on Tg (and thus we can identify K with a subsetof Mg ). All of our implied constants will depend on K .

NotationSuppose that W �Q1Tg and that s > 0. Let W.s/ denote the set of q 2Q1Tg suchthat there exists q0 2W on the same leaf of F uu as q such that dH .q; q0/ < s. Recallthat, for a subset A � Tg , Nbhdr.A/ denotes the set of points within Teichmüllerdistance r of A. We will also use A.r/ to denote Nbhdr.A/. Let N and C be asin Section 2.2 so that, for a set F contained in a leaf of F uu, we have N.C.F //D�˛uu.F /.

LEMMA 4.1Suppose that U � ��1.K/, ı > 0, and t > 0. Let W D gtU \ ��1.K/. There is aC.ı/ > 0 so that

m�Nbhd2.�.W //

� C.ı/

�C.W.ı//

�(4.1)

(by our convention, the constant C.ı/ depends on K as well). Also,

�C.W.1//

� C.ı/

�C.W.ı//

�: (4.2)


ProofLet ı0 D ı0.K; ı/ be a constant to be chosen later. We decompose U into pieces U˛such that each piece is within the (modified) Hodge distance ı0=2 of a single leaf ofF uu. Let

W˛ D gtU˛ \ ��1.K/:

Then

m�Nbhd2.�.W //

�X˛

m�Nbhd2.�.W˛//

�:

In view of Theorem 3.10, the number of pieces is bounded depending only on K andı (since ı0 D ı0.K; ı/). Thus, it is enough to show that (4.1) holds with W replacedby W˛ .

We now claim that without loss of generality we may assume that U˛ has thefollowing product property: given q1, q2 2 U˛ , there exists q02 2 U˛ on the same leafof F uu as q1 and on the same leaf of F s as q2. If not, let

U 0˛ DU˛ [®F uu.q1/\F s.q2/ W q1; q2 2 U˛

¯:

Then C.U˛/D C.U 0˛/, and therefore C.gtU˛/D C.gtU 0˛/. Also, if ı0 D ı0.ı;K/is sufficiently small, then W 0˛ D gtU

0˛ \ ��

�1.K/ satisfies W 0˛.ı0/�W˛.ı/. There-

fore, we can proceed with the rest of the proof with ı0 instead of ı and with U 0˛ insteadof U˛ . Therefore, without loss of generality, we may assume that U˛ has the productproperty. Therefore, gtU˛ also has the product property.

Pick a maximal � �.W˛/ such that, for any two distinct X;Y 2 , dT .X;Y /D 1. Then,

Nbhd2��.W˛/

��[X2�

B .X; 3/;

and hence,

m�Nbhd2.�.W˛//

�m

�[X2�

B .X; 3/�XX2�

m�B.X;3/

� C.K/j j; (4.3)

where j j denotes the cardinality of , and we have used the fact that � �K .For each X 2 , q 2W˛ \ ��1.X/. Let 0 �W˛ denote the resulting set of q.

Let BuuE .q; r/ denote the set of q0 2Q1Tg on the same leaf of F uu as q with dE .q;q0/ < r . We claim that, for ı0 sufficiently small, we can pick ı2 depending only on K

such that for all distinct pairs q1; q2 2 0, we have

C�BuuE .q1; ı2/

�\ C

�BuuE .q2; ı2/

�D;: (4.4)


To prove (4.4), suppose that q1; q2 2 0 and that q1 ¤ q2. Let q02 be such that q1and q02 are on the same leaf of F uu and such that q2 and q02 are on the same leaf ofF s . By Theorem 3.15, we have dH .q2; q02/ < cH ı0. Therefore, by Theorem 3.10 andLemma 3.9, we can choose ı0 small enough so that dT .�.q2/;�.q02// 1=5. Hence,�.q02/� �K 0, where K 0 � Tg is compact, and by the triangle inequality,

dT��.q1/;�.q

02/�� dT

��.q1/;�.q2/

�� dT

��.q2/;�.q

02/�� 4=5: (4.5)

By Lemma 3.9, there exists ı1 > 0 (depending only on K 0) such that, for all q, q0 2K 0 on the same leaf of F uu with dE .q; q0/ < ı1, we have dT .�.q/;�.q0// < 1=5.Then, by (4.5) and the triangle inequality, BuuE .q1; ı1/ and BuuE .q02; ı1/ are disjoint.It follows that as subsets of P MF ,

C�BuuE .q1; ı1/

�\ C

�BuuE .q02; ı1/

�D;: (4.6)

Let F s.q/ denote the leaf of F s through q. Since C is continuous and K 0 is com-pact, there exist constants 0 < ı2 < ı1 and ı3 > 0 (depending only on K 0) such that,for all q; q0 2K 0 with q 2 F s.q0/ and dH .q; q0/ < ı3, we have, as subsets of P MF ,

C�BuuE .q; ı2/

�� C

�BuuE .q0; ı1/

�:

We now choose ı0 < ı3=cH . Then, since q2 and q02 are on the same leaf of F s andsince dH .q2; q02/ < cH ı0 < ı3, we have

C�BuuE .q2; ı2/

�� C

�BuuE .q02; ı1/

�:

Now using (4.6), we get that, as subsets of P MF ,

C�BuuE .q1; ı2/

�\ C

�BuuE .q2; ı2/

�� C

�BuuE .q1; ı1/

�\ C

�BuuE .q02; ı1/

�D;:

This completes the proof of (4.4).By Theorem 3.10, there exists 0 < ı3 < ı2 such that, for all q 2 0 and all q0 2

F uu.q/ with dE .q; q0/ < ı3, we have dH .q; q0/ < ı. For q 2 0, let

H.q/D C�BE .q; ı3/

�� C

�W.ı/

�:

Consider the collection of balls ¹H.q/ W q 2 0º. By (4.4), the sets H.q/ are pairwisedisjoint viewed as subsets of P MF (or, alternatively, the subsets Cone.H.q// �MF intersect only at the origin). Also, by the definition of the Thurston measure and the compactness of K , there exists a constant c D c.K; ı/ such that

N�H.q/

�� c for all q 2 0.

Hence,


�C.W.ı//

��

� [q2�0

H.q/�DXq2�0

�H.q/

�� cj 0j D cj j: (4.7)

Now (4.1) (with W replaced by W˛) follows from (4.3) and (4.7). Finally,

�C.W.1//

� C.ı;K/

�C.W.ı//

�;

since dH is equivalent to dE . �

The sets Ki and Ui . Let K1 � Q1Mg be a compact set. (In our application,K1 will be chosen disjoint from the multiple zero locus.) Let K3 � K2 � K1 and1 � ı > 0 be such that if q 2Ki and dH .q; q0/ < cH ı, then q0 2Ki�1, where cH isas in Theorem 3.15(a). We assume that �.K3/ > .1=2/, where � is the normalizedLebesgue measure on Q1Mg . For T0 > 0, let Ui D Ui .T0/ be the set of q 2Q1Mg

such that there exists T > T0 so thatˇ̌®t 2 Œ0; T � W gtq 2K

ci

¯ˇ̌� .1=2/T:

Then, for all T > T0 and all q … Ui ,ˇ̌®t 2 Œ0; T � W gtq 2K

ci

¯ˇ̌< .1=2/T: (4.8)

From the definition, we have U1 � U2 � U3. By the ergodicity of the geodesic flow,for every � > 0 there exists T0 > 0 such that �.U3/ < � .

Let K1 DK . We can choose compact subsets K0, K2, K3 of Tg such that, for0 i 2, for any q 2 ��1.Ki /, and for any q0 on the same leaf of F uu as q withdH .q; q

0/ < cH , we have q0 2 ��1.KiC1/. By (the lower bound in) Theorem 3.10and Lemma 3.9, each Ki is compact. Let U 0i D p

�1.Ui /\ ��1.Ki /, where p is the

natural map from Q1Tg to Q1Mg . Note that U 01 �U02 �U

03 �Q1Tg .

LEMMA 4.2Let U 0i , 1 i 3 be as in the preceding paragraph. Then, for all t > 0,

m�Nbhd2.�.gtU

01/\ �K/

� C.ı/eht

�C.U 02/

�;

and if W.1/ is defined as in Lemma 4.1 with W D gtU 01 \ ��1.�K/, then

�C.W.1//

� C 0.ı/eht

�C.U 02/

�:

In particular (see Theorem 2.2), for any � > 0, it is possible to choose T0 such that, ifU1 is defined by (4.8) and if U 01 D U1 \ �

�1.K1/, then for all t > T0, we have

m�Nbhd2.�.gtU

01//\ �K

� �eht ; (4.9)


and for W D gtU 01 \ ��1.�K/, we have

�C.W.1//

� �eht : (4.10)

ProofWe apply Lemma 4.1 to the setW D gtU 01\�

�1.�K/. We claim thatW.ı/� gtU 02.Indeed, suppose that gtq0 2W.ı/. Then there exists gtq 2W with

dH .gtq;gtq0/ < ı:

Then, by Theorem 3.15(a), for all 0 s t ,

dH .gsq;gsq0/ < cH ı: (4.11)

Since W D gtU 01, gtq 2W implies that q 2 U 01. Then, assuming that t > T0, for atleast half the values of s 2 Œ0; t �,

gsq …K1: (4.12)

Then, (4.11) and the definition of K2 imply that, for the s 2 Œ0; t � for which (4.12)holds, gsq0 … K2. This implies that q0 2 U2. Since dH .q; q0/ < cH ı < cH and q 2��1.K/D ��1.K1/, we have q0 2 ��1.K2/. Thus, q0 2 U2 \ ��1.K2/D U

02, and

so gtq0 2 gtU2. This implies the claim and thus the first two statements of the lemma.The same argument as the proof of the claim shows that if q 2 U 02 and q0 2

F ss.q/ \ ��1.K/ with dH .q; q0/ < ı, then q0 2 U 03. This (together with Theo-

rem 3.10) implies that there exists C1.ı/ such that

�C.U 02/

� C1.ı/�.U

03/:

Hence if we choose T0 > 0 so that�.U3/ < C.ı/C1.ı/�, then (4.9) follows. Similarly,if we choose T0 > 0 so that in addition �.U3/ < C 0.ı/C1.ı/�, then (4.10) follows. �

Proof of Theorem 2.7Let K 0 D K1, and let T0, U1, and U 01 be as in Lemma 4.2. Then, for R > T0 andX 2K ,

BR.X;K;K 0/�[

0�t�R

�.gtU01/\ �K �

bRc[nD0

[t2Œn;nC1

�.gtU01/\ �K;

where bxc denotes the integer part of x. Then,

m�Nbhd1.BR.X;K;K 0//

�

bRcXnD0

m�Nbhd2.�.gnU

01/\ �K/

� C�

bRcXnD0

ehn;

where we have used (4.9). Since � is arbitrary, Theorem 2.7 follows. �


LEMMA 4.3Suppose that V � ��1.K/ and that ı0 > 0. Then, for t sufficiently large (dependingon K , V , and ı0),

m�Nbhd1.�.gtV /\ �K0/

� C

�C.V .ı0//

�eht ;

where C depends only on K .

ProofLet Y D gtV \ �K0. Choose T0 so that (4.10) holds for t > T0 and .C.V .ı0///instead of �. Let U1, U 01, and W D gtU 01 \�

�1.K/ be as in Lemma 4.2, so in partic-ular, for t > T0,

�C.W.1//

�

�C.V .ı0//

�eht : (4.13)

We claim that there exist T1 > T0, depending only on K , such that for t > T1,

Y.1/DW.1/[ gt�V.ı0/

�: (4.14)

Indeed, if gtq 2 Y.1/, then by definition there exists q0 2 V �K with dH .gtq;gtq0/ < 1, and q0 is on the same leaf of F uu as q. We consider two cases: either q0 2

U1 or q0 … U1. If q0 2 U1, then q 2 U 01 D U1 \K ; hence gtq0 2 gtU 01 \ �K DW .Hence in this case, gtq 2W.1/. If q0 … U1, then by (4.8) and Theorem 3.15(b),

dH .q; q0/D dH .g�tgtq;g�tgtq

0/ Ce�ctdH .gtq;gtq0/ Ce�ct :

We choose T1 > T0 so that Ce�cT1 < ı0. Thus, in this case, for t > T1, q 2 V.ı0/, andhence (4.14) follows.

Now, for t > T1,

�C.Y.1//

�

�C.W.1//

�C

�C.gt .V .ı

0///� 2

�C.V .ı0//

�eht ;

where we have used (4.13). We now apply Lemma 4.1 with ıD 1. The lemma followsfrom (4.1). �

Proof of Theorem 2.6This follows immediately from Lemma 4.3 and the fact that we can choose a relativelyopen V �K and ı0 > 0 such that V contains the intersection of K with the multiplezero locus, and

�C.V .ı0//

�is arbitrarily small (see Theorem 2.2). �

5. Volume asymptoticsIn this section we prove Theorem 1.3. Let FR.X;Y / D j� � Y \ BR.X/j be as inSection 2.6. We need the following.


THEOREM 5.1Given X 2 Tg , there exists C D C.X/ such that, for all Y 2 Tg ,

FR.X;Y / C.X/ehR:

Notation� Let Cg.N/ be the set of isotopy classes of integral multicurves on a surface of

genus g:� Given X;Y 2 Tg , define

MR.X;Y /D®� � Y; � 2 �

ˇ̌� � Y 2BR.X/

¯so that FR.X;Y /D #.MR.X;Y //:

Proof of Theorem 1.3, assuming Theorem 5.1By definition of FR,

m�BR.X/

�D

ZMg

FR.X;Y /dm.Y /:

We multiply both sides by e�hR and take the limit as R!1. By Theorem 5.1, wecan apply the bounded convergence theorem to take the limit inside the integral. Nowthe theorem follows from Theorem 1.2. �

A similar argument yields the following.

THEOREM 5.2Suppose that X 2 Tg and that U� S.X/. Then, as R!1,

m�BR.X/\ SectU.X/

��

1

hm.Mg/ehRƒ.Y /

ZU

��.q/dsX .q/:

In the rest of this section we prove Theorem 5.1. Along the way, we prove Theo-rem 2.4.

Estimating extremal lengths. Consider the Dehn–Thurston parameterization (see[PH]) of the set of multicurves

DT W Cg.N/! .ZC �Z/3g�3

defined by

DT.ˇ/D�i.ˇ;˛i /; tw.ˇ;˛i /

�3g�3iD1

;


where i.�; �/ denotes the geometric intersection number and where tw.ˇ;˛i / is thetwisting parameter of ˇ around ˛i (see [PH] for more details). By a theorem of Bers,we can choose a constant Cg depending only on g such that for any surface Y 2 Tg ,there exists a pants decomposition P D ¹˛1; : : : ; ˛3g�3º on Y such that, for 1 i 3g � 3,

Ext˛i .Y / C2g :

We call such a pants decomposition a bounded pants decomposition for Y: The fol-lowing result is proved by Minsky.

THEOREM 5.3 ([Mi2, Theorem 5.1 and (4.3)])Suppose that Y 2 Tg , and let P D P .Y /D ¹˛1; : : : ; ˛3g�3º be any bounded pantsdecomposition on Y . Then, given a simple closed curve ˇ, Extˇ .Y / is bounded fromabove and below by

max1�j�3g�3

h i.ˇ;˛j /2Ext˛j .Y /

C tw2.ˇ;˛j /Ext˛j .Y /i

(5.1)

up to a multiplicative constant depending only on g.

Theorem 5.3 gives a bound on the Dehn–Thurston coordinates of a simple closedcurve in terms of its extremal length.

RemarkThe definition of the twist used in [Mi2, (4.3)] is different from the definition we areusing here. We follow the definition used in [PH, p. 15]. Given a connected simpleclosed curve ˛ on †g , let h˛ 2 � denote the right Dehn twist around ˛. Then, interms of our notation,

tw�hr˛.ˇ/; ˛

�D tw.ˇ;˛/C r � i.ˇ;˛/: (5.2)

COROLLARY 5.1Let ˛D ¹˛1; : : : ; ˛3g�3º be a bounded pants decomposition of X 2 Tg . Then there isa constant c1 > 0 such that, for any simple closed curve ˇ on †g ,

tw.ˇ;˛i / c1 �

pExtˇ .X/pExt˛i .X/

and

i.ˇ;˛i /q

Extˇ .X/ �q

Ext˛i .X/ c1q

Extˇ .X/:


Estimating the number of multicurves. Define

E.Y;L/D #ˇ̌®˛ 2 Cg.N/

ˇ̌ pExt˛.Y /L

¯ˇ̌:

Fix �0 > 0 small enough such that, if ˛ and ˇ on Y 2 Tg satisfy Ext˛.Y / �20 andExtˇ .Y / C 2g , then i.˛;ˇ/D 0: Note that any bounded pants decomposition P onY includes all simple closed curves of extremal length �20 on Y .

Let

G.Y /D 1CY

Ext� .Y /��20

1pExt� .Y /

;

and assume that the product ranges over all simple closed curves � on the surface Ywith Ext� .Y / �20 .

Using Theorem 5.3, we obtain the following.

THEOREM 5.4There exists a constant C > 0 such that, for every Y 2 Tg and L> 0, we have

E.Y;L/ C �G.Y / �L6g�6:

Moreover, for any Y , if 1=L is bounded by an absolute constant times the square rootof the extremal length of the shortest curve on Y , then

E.Y;L/ CL6g�6:

Sketch of the proofIn order to use the bound given by (5.1), first we fix a bounded pants decomposition

P D ¹˛1; : : : ; ˛3g�3º

on Y . Note that this pants decomposition should include all small closed curveson Y: Let mi D i.ˇ;˛i /, let ti D tw.ˇ;˛i /, and let si D

pExt˛i .Y /. So by (5.1),

Extˇ .Y / L2 implies that misiC jti jsi D O.L/. We use the following elementary

lemma.

LEMMA 5.5For s > 0, define As.L/ by

As.L/D°.a; b/

ˇ̌̌a; b 2 ZC; a � sC

b

sL

±� ZC �ZC:

Then for any L > 0, jAs.L/j 4max¹s; 1=sº � L2: For L > max¹s; 1sº, we have

As.L/ 4L2:


Now applying Lemma 5.5 to the si and using the Dehn–Thurston parameteriza-tion of multicurves, we get

E.Y;L/

3g�3YiD1

jAsi .L/j CG.Y /L6g�6:

�

Proof of Theorem 2.4This follows from the second part of Theorem 5.4. Recall (see Section 2.1) that theextremal length can be extended continuously to a map

Ext WMF � Tg !RC

such that Extt.X/ D t2Ext.X/: Also the space MF has a piecewise linear inte-gral structure, and elements of Sg are in one-to-one correspondence with the integralpoints. Hence,

ƒ.X/D Vol® 2MF

ˇ̌ qExt�.X/ 1

¯D

Vol¹ 2MF jp

Ext�.X/Lº

L6g�6D limL!1

E.x;L/

L6g�6; (5.3)

where to justify the last equality we use the following fact: as L!1, the number oflattice points in the dilation LA is asymptotic to the volume of LA, where AD ¹ 2MF j

pExt�.X/ 1º.

On the other hand, by the second assertion in Theorem 5.4, whenL is big enough,E.Y;L/ CL6g�6. Therefore, ƒ.Y / C; where the bound does not depend on Y:

�

RemarkLet p W Tg !Mg be the natural projection. Fix a compact subset K �Mg . Forsimplicity, we will also denote p�1.K/� Tg by K . If P D ¹˛iº is a bounded pantsdecomposition on X 2K , then for any 1 i 3g � 3,

1

CK

q

Ext˛i .X/ CK ;

where CK is a constant which only depends on K: Therefore, by Corollary 5.1, forthe elements in the thick part of the Teichmüller space, the Dehn–Thurston coordi-nates of a simple closed curve are bounded by its extremal length. More precisely,if ˛ D ¹˛1; : : : ; ˛3g�3º is a bounded pants decomposition of X 2K , then there isa constant c1 > 0, depending only on K , such that for any simple closed curve ˇon †g ,


tw.ˇ;˛i / c1 �q

Extˇ .X/; i.ˇ;˛i / c1 �q

Extˇ .X/:

PROPOSITION 5.6There exists a constant C3 depending only on K such that for every X 2K , Y 2 Tg ,and R > 0, we have

FR.X;Y / C3E.Y; eR/:

Our goal is to assign to any point Z D � � Y 2MR.X;Y / a unique integral mul-ticurve ˇZ 2E.Y; eR/: This would imply that

FR.X;Y / Ce.6g�6/R:

(Note that we are assuming that X is in the compact part of Tg and that d.X;Y / < R;this implies that the shortest curve on Y has extremal length at least constant timese�2R, hence we may use the second statement of Theorem 5.4.)

The most natural candidate for ˇZ is ��1˛, where ˛ is a fixed pants decompo-sition of X . However, this correspondence is not one-to-one. Therefore, we need tomodify the construction.

Given Z D � � Y 2MR.X;Y / and ˛ 2 Sg , we havepExt˛.Z/pExt˛.X/

eR:

So by setting LD eR and ˇD ��1.˛/, we haveqExtˇ .Y /L �

pExt˛.X/: (5.4)

Given Z D � �Y 2MR.X;Y /, let ˛.Z/i D ��1˛i 2 Sg : Then from (5.4), we getqExt˛.Z/i .Y / CK �L:

Moreover, forZ1 D �1Y;Z2 D �2Y 2 � �Y , we have ˛.Z1/D ˛.Z2/ if and onlyif there are r1; : : : ; r3g�3 2 Z such that

�1 D hr1˛1� � �h

r3g�3˛3g�3 � �2:

Moreover, we have the following.

LEMMA 5.7Let ˛ D ¹˛1; : : : ; ˛3g�3º be a bounded pants decomposition on X , and suppose thatY0 2BR.X/. If hr1˛1 � � �h

r3g�3˛3g�3.Y0/ 2BR.X/, then for 1 i 3g � 3,


jri j �q

Ext˛i .Y0/ C2eRp

Ext˛i .X/:

Here C2 is a constant independent of X and Y0:

Sketch of the proof of Lemma 5.7Let si D

pExt˛i .Y0/; and let LD eR. The key point in the proof is in [Mi2, Lem-

ma 6.3]. From this lemma, for any i there exists a multicurve ˇi of extremal lengthb2i D Extˇi .Y0/ such that

i.ˇi ; ˛i /� csi � bi :

Since d.X;Y0/R,p

Extˇi .X/ bi �L: Also, the assumption

d�h�r1˛1� � �h

�r3g�3˛3g�3 .X/;Y0

�R

implies that

qExt

hr1˛1��hr3g�3˛3g�3

.ˇi /.X/D

qExtˇi

�h�r1˛1 � � �h

�r3g�3˛3g�3 .X/

� bi �L: (5.5)

Then we have the following.(1) Since

pExtˇi .X/ bi �L; Corollary 5.1 implies that the twist and intersection

coordinates of the curve ˇi are bounded by a multiple of bi �L; in particular,

j tw.ˇi ; ˛i /j DO� bi �Lp

Ext˛i .X/

�:

(2) Similarly, by equation (5.5), applying Corollary 5.1 for hr1˛1 � � �hr3g�3˛3g�3.ˇi / on

X with respect to ¹˛iº implies that

ˇ̌tw�hr1˛1 � � �h

r3g�3˛3g�3.ˇi /; ˛i

�ˇ̌DO

� bi �LpExt˛i .X/

�:

(3) On the other hand, by the definition (see (5.2)), we haveˇ̌tw�hr1˛1 � � �h

r3g�3˛3g�3.ˇi /; ˛i

�ˇ̌D jri � i.˛i ; ˇi /C tw.ˇi ; ˛i /j

� jri j � i.˛j ; ˇi /� j tw.ˇi ; ˛i /j:

So we have

jri j � si � bi 1

cjri j � i.ˇi ; ˛i / C2L � bi H) jri j � si

C2LpExt˛i .X/

: �


RemarkAs a result, if the assumption of Lemma 5.7 holds and if X 2K , then

jri j �q

Ext˛i .Y0/ CeR;

where C is a constant which only depends on K:

We remark that for any two disjoint simple closed curves 1 and 2, and m1;m2 > 0, we have, by the definition of extremal length,q

Extm1��1.X/Cq

Extm2��2.X/�q

Extm1��1Cm2��2.X/:

COROLLARY 5.2If the assumption of Lemma 5.7 holds and X 2K , then for b̨DP3g�3

iD1 jri j � ˛i , wehave p

Extb̨.Y0/ C3eR:

Therefore, b̨ also defines a multicurve of extremal length bounded by e2R. Thiscompletes the proof of Proposition 5.6, and thus, in view of Theorem 5.4, the proofof Theorem 5.1. �

Appendix. Proof of Theorem 2.2In this section, Q�Tg denotes the space of nonzero quadratic differentials on markedcompact surfaces of genus g, without restriction on area.

A local coordinate system near the multiple zero locus. Near the multiple zerolocus, the coordinate system given by the period map (see Lemma 3.8) is singular.Instead, we use an alternative coordinate system from [HMa].

Suppose that q0 2Q�Tg has multiple zeros, say, w1; : : : ;wm. Let mi be the mul-tiplicity of q0 at wi . Then by [HMa, Proposition 3.1], for 1 i m, there exists achoice of local coordinate zi , mapping wi to zero, such that any q 2Q�Tg near q0,on a neighborhood of wi , has the form

q D�zmii C

mi�2XjD0

aij zji

�.dzi /

2:

The coordinates zi are uniquely determined by q. Fix some ı > 0, and let w0i bethe point corresponding to zi D ı. For q 2Q�Tg near q0, let QX denote the canonicaldouble cover which makes the foliations corresponding to q orientable, and let �be the involution of QX so that QX=� is the surface X (with the flat structure givenby q). Let †ı D ��1.w01; : : : ;w

0m/, and let H denote the relative homology group


H1. QX;†ı ;Z/. Let Hodd denote the odd part of H under the action of the involution� . Let �1; : : : ; �m be a integral basis for Hodd. Let

�i D

Z�i

pq:

Suppose thatmi is even. Let wij , 1 j mi denote the zeros of q which tend towi as q! q0. Let i be a fixed small circle in the coordinates zi , centered at zi D 0(i.e., wi ). Then if q is sufficiently close to q0, i separates all the wij from the rest ofthe surface. Let bi D

R�i

pq. Then, a residue calculation shows that

bi D ami2 �1C some polynomial in aij ; mi � 2� j �

mi

2: (A.1)

PROPOSITION A.1 (see [HMa])If q0 is not a square of an abelian differential, then there exists a neighborhood U �Q�Tg of q0 such that the aij and the �i are smooth local coordinates on U . If q0 isthe square of an abelian differential, then there exists a neighborhood U of q0 suchthat under the constraint

Pi bi D 0, the aij and the �i are local coordinates on U .

In both cases, near q0, the natural projection map � WQ�Tg ! Tg is a submer-sion in these coordinates.

ProofSee [HMa, Proposition 4.7]. �

The constraintPi bi D 0 appears because in the case when q0 is the square of

an abelian differential, the part of the surface which is outside all the circles i hasan orientable foliation. In that case, in view of (A.1), we may drop ami

2 �1for some i

from the coordinate system.We introduce a covering of Q�Tg by open sets U˛ such that each U˛ is the

neighborhood U of Proposition A.1 for some q0 2 Q�Tg . We may assume that theU˛ are invariant under the operation of multiplying the quadratic differential by a realnumber. We also assume that the covering is uniformly locally finite on compact sets,that is, that for any compact set K 2 Tg , there exists a number N depending only onK such that each point in O��1.K/ belongs to at most N sets U˛ .

Let ˛ be a partition of unity subordinate to U˛ . For q; q0 2 U˛ , let ¹aij , �iºbe the coordinates of q, and let ¹a0ij , �0iº be the coordinates of q0. Let D˛.q; q0/DPij jaij � a

0ij j C

Pi j�i � �

0i j. For q and q0 sufficiently close so that they belong to

the same U˛ , let D.q; q0/DP˛ ˛D˛.q; q

0/.Recall that dT .X;Y / denotes the Teichmüller distance betweenX and Y and that

S.X/ denotes the sphere of unit area quadratic differentials which are holomorphicat X .


LEMMA A.2Suppose that X , Y in Tg are sufficiently close and belong to a compact set K . Then,there exist c > 1 depending only on K such that

c�1 infq2S.X/

D�q;��1.Y /

� dT .X;Y / c sup

q2S.X/

D�q;��1.Y /

�: (A.2)

ProofLet K be the pullback of K from Tg to Q1Tg . Then K is also compact. Near anyq 2K , by Proposition A.1, the map � W Q�Tg ! Tg is a submersion, and thereforecan be approximated by a linear map, for which (A.2) is immediate. The rest followsby compactness. �

Let ˇ˛ be the measure on U˛ � Q�Tg given byQi;j daij

Qi d�i . Let ˇ DP

˛ ˛ˇ˛ .For V � ��1.X/, let V � � Q�Tg denote the set of all quadratic differentials

which have the same horizontal foliation as some q 2 V .

LEMMA A.3For every ı > 0, any X 2 Tg , and any q0 2Q�Tg , there exists an open subset V of��1.X/ containing q0, such that for all sufficiently small � > 0,

ˇ�V � \ ��1.B.X; �//

�< ı�6g�6:

ProofFix ı0 > 0. Recall that by a horosphere (e.g., leaf of F ss) in Q�Tg we mean theset of quadratic differentials with a fixed horizontal foliation. Since the foliation byhorospheres is continuous, there exists a neighborhood V of q0 2 ��1.X/ such thatV �BD.q0; ı

0/, whereBD denotes a ball in the metricD. Then, for � > 0 sufficientlysmall, and in view of Lemma A.2, there exist constants C1, C2 such that

��1�V � \B.X; �/

��®q 2Q�Tg WD

�q;BD.q0;C2ı

0/\ ��1.X/�<C1�

¯: (A.3)

Since ��1.X/ is smooth, the ˇ-measure of the set on the right-hand side of (A.3) is�6g�6c.ı0/, where c.ı0/! 0 as ı0! 0. This implies the lemma. �

PROPOSITION A.4The measure � on Q�Tg is absolutely continuous with respect to ˇ, and for anycompact subset K of Tg there exists a constant C such that for all q 2 ��1.K/,ˇ̌d�dˇ.q/ˇ̌< C . Also, away from the multiple zero locus,

ˇ̌d�dˇ.q/ˇ̌

is a smooth nonvan-ishing function of q.


RemarkIt is also possible to show that there exists a constant C 0 depending only on K suchthat for all q 2 ��1.K/, one has C 0 <

ˇ̌d�dˇ.q/ˇ̌. Since we do not need this, we omit

the proof.

Proof of Theorem 2.2, assuming Proposition A.4By Lemma A.2, ˇ

��1.B.X; �//

�DO.�6g�6/. Therefore, by Proposition A.4, there

exist 0 < c1 < c2 such that

c1�6g�6 <�

��1.B.X; �//

�< c2�

6g�6: (A.4)

Now,

dsX .V /D lim�!0

��V � \ ��1.B.X; �//

��1.B.X; �//

� lim sup

�!0

Cˇ�V � \ ��1.B.X; �//

��1.B.X; �//

� Cı�6g�6

c1�6g�6;

where for the first inequality we used Proposition A.4, and for the last estimate weused Lemma A.3 and (A.4). Since ı is arbitrary, the theorem follows. �

The rest of the section consists of the proof of Proposition A.4. To simplify nota-tion, we work with one zero of q0 at a time. We write

zmC

m�1XiD0

aizi D

mYjD1

.z � zj /:

Then the ai are symmetric polynomials in the zj . It is well known that the Jacobianof the map from the zeros’ zj to the coefficients’ ai is the Vandermonde determinant.In other words, if we use the notation @.y1;:::;yn/

@.x1;:::;xn/for the Jacobian determinant of the

matrix®@yi@xj

¯1�i;j�n

, we have

@.a0; : : : ; am�1/

@.z1; : : : ; zm/DYi<j

.zi � zj /: (A.5)

(This follows from comparing degrees and noting the antisymmetry of the determi-nant.)

We choose a basis for the homology relative to the zeros. This amounts to choos-ing a spanning subtree T from the complete graph connecting the zeros’ zj . We maychoose the tree in such a way that for any zi and zj , jzi �zj j is within a multiplicativeconstant (depending only on m) of the length of the path in T connecting zi and zj .


Let the (oriented) edges of T be e1; : : : ; em�1. Let eCk

be the zero at the head of ek ,and let e�

kbe the zero at the tail of ek . We write Eek D e

Ck� e�

k(so Eek 2C).

In our setting, we need to restrict to am�1 D 0. Note that am�1 D z1C � � � C zm.It follows that

@.a0; : : : ; am�2/

@.Ee1; : : : ; Eem�1/D

@.a0; : : : ; am�1/

@.Ee1; : : : ; Eem�1; am�1/

[email protected]; : : : ; am�1/

@.z1; : : : ; zm/

@.z1; : : : ; zm/

@.Ee1; : : : ; Eem�1; am�1/

DYi<j

.zi � zj /; (A.6)

where we have used (A.5).If zj is a zero and if e is an edge of T , let

dC.zj ; e/Dmax.jzj � e�j; jzj � e

Cj/:

We need the following combinatorial lemma.

LEMMA A.5We have

Ye2T

mYpD1

dC.zp; e/1=2 C

Yi<j

jzi � zj j; (A.7)

where C depends only on m.

ProofLet e0 be the longest edge of T . If we cut along e0, we separate the tree into two sub-trees T1 and T2, say, of sizem1 andm2. If zi 2 T1 and zj 2 T2, then by the assumptionon T , jzi � zj j is comparable to the length of the path in the tree connecting zi to zj .This path contains e0, which is by assumption the longest edge in the tree. Therefore,if zi 2 T1 and zj 2 T2, then jzi � zj j is within a multiplicative constant of e0. Hence,the left-hand side of (A.7) is within a multiplicative constant of

je0jm1m2Yi<j

zi ;zj2T1

jzi � zj jYi<j

zi ;zj2T2

jzi � zj j:

To estimate the right-hand side of (A.7), note that each factor of the form dC.zp; e/1=2

(where zp 2 T1 and either e 2 T2 or e D e0) is within a multiplicative constant ofje0j1=2. The number of such factors ism1m2. Also, the same factors appear when zp 2


T2 and e 2 T1 or e D e0. Then the right-hand side of (A.7) is within a multiplicativeconstant of

je0jm1m2Ye2T1

Yzp2T1

dC.zp; e/1=2

Ye2T2

Yzp2T2

dC.zp; e/1=2:

The estimate (A.7) now follows by induction. �

LEMMA A.6Suppose that z1; : : : ; zm are in some bounded set K . Let

�k D

Z eC

k

e�k

p.z � z1/ � � � .z � zn/dz

so that �k is the holonomy of the edge ek . Thenˇ̌̌@.�1; : : : ;�n�1/@.Ee1; : : : ; Eem�1/

ˇ̌̌ C

Yi<j

jzi � zj j;

where C depends only on m and K .

ProofSuppose for the moment that z1; : : : ; zm are independent variables. We claim that

ˇ̌̌@�k@zj

ˇ̌̌ C

nYpD1

dC.zp; ek/1=2; (A.8)

where C depends only on m and K . Indeed,

@�k

@zjD

Z eC

k

e�k

p.z � z1/ � � � .z � zn/

2.z � zj /dz:

If we write the numerator in the integral as P.z � zj /1=2, then P is bounded by aconstant times

QnpD1;p¤j

dC.zp; ek/1=2. Then, the integral is bounded by

jP j

2

ˇ̌̌Z eC

k

e�k

dz

.z � zj /1=2

ˇ̌̌:

If dC.zj ; ek/ >12jeCk� e�

kj, then

ˇ̌̌Z eC

k

e�k

dz

.z � zj /1=2

ˇ̌̌ C

jeCk� e�

kj

dC.zj ; ek/1=2 C 0jeC

k� e�k j

1=2


(where we have used the assumption that jeCk� e�

kj is bounded). If dC.zj ; ek/ <

12jeCk� e�

kj, then

ˇ̌̌Z eC

k

e�k

dz

.z � zj /1=2

ˇ̌̌ C 00jeC

k� e�k j

1=2;

where C 00 depends only on m and K . Thus,

ˇ̌̌@�k@zj

ˇ̌̌ C jeC

k� e�k j

1=2Yp¤j

dC.zp; ek/1=2:

But by the choice of the tree T , for any j , dC.zj ; ek/� cjeCk� e�

kj, where c depends

only on m. Thus, (A.8) holds.Since

PmjD1 zj D 0, there are m � 1 linearly independent zj , and therefore we

can express zj DPm�1kD1 jk Eek , where the jk are bounded depending only on m.

Then, we get

ˇ̌̌@�j@Eek

ˇ̌̌ C 0

mYpD1

dC.zp; ek/1=2:

Lemma A.6 now follows from Lemma A.5. �

Proof of Proposition A.4It is enough to show that for any ˛, j @�

@ˇ˛j C on U˛ . As above, we work with one

zero of q0 at a time. By (A.6) and Lemma A.6,

ˇ̌̌@.�1; : : : ;�m�1/@.a0; : : : ; am�2/

ˇ̌̌Dˇ̌̌@.�1; : : : ;�m�1/@.Ee1; : : : ; Eem�1/

ˇ̌̌ˇ̌̌ @.Ee1; : : : ; Eem�1/@.a0; : : : ; am�2/

ˇ̌̌ C; (A.9)

where C depends only on m and K .Let �0 D d�1 d N�1; : : : ; d�m�1 d N�m�1, and let ˇ0 D da0 d Na0; : : : ;

dam�2 d Nam�2. Then, by (A.9),

d�0

dˇ0Dˇ̌̌@.�1; : : : ;�m�1/@.a0; : : : ; am�2/

ˇ̌̌2 C 2:

Now,

d�

dˇ˛DYj

d�0j

dˇ0j;

where the product is over distinct zeros of q0. Now Proposition A.4 followsfrom (A.9). �


Acknowledgments. The authors are grateful to Giovanni Forni, Vadim Kaimanovich,Yair Minsky, and Kasra Rafi for useful discussions, and especially to Howard Masurfor his help with all parts of the paper, in particular the appendix. We are also gratefulto the Institute for Advanced Study, the Institut des Hautes Études Scientifiques, andthe Mathematical Sciences Research Institute for their support.

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Athreya

Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA;

[email protected]

Bufetov

Steklov Institute of Mathematics, Russian Academy of Sciences, 119991, Moscow, Russia;

[email protected]

Institute for Information Transmission Problems, Moscow, 127994, Russia

National Research University Higher School of Economics, Moscow, 117312, Russia















mailto:[email protected]



Current: Department of Mathematics, Rice University, Houston, Texas 77251, USA;

[email protected]

Eskin

Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA;

[email protected]

Mirzakhani

Department of Mathematics, Stanford University, Stanford, California 94305, USA;

[email protected]




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Lattice point asymptotics and volume growth on Teichmüller space

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