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EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS ON

HYPERBOLIC SURFACES

ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI

Abstract.We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at mostLon a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmuller geodesic ow.

1.Introduction

Letg2, and letSbe a compact Riemann surface of genusg. LetT(S) be the Teichmuller space of complete hyperbolic metrics onS, and let

M(S) =T(S)=Modg

be the corresponding moduli space, where Mod gis the mapping class group ofS. LetM2 M(S). Problems related to the asymptotic growth rate of the number of closed geodesics onMhave been long studied. In particular, thanks to works of Delsart, Huber, and Selberg we have the following: There exists some=(M)>0 so that the number of closed geodesics of length at mostLonMequals (1) Li(eL) +OM(eL); where Li(x) =Rx

2dtlogt; see [Bus] and references there.

More generally, the growth rate of the number of closed geodesics on a negatively curved compact manifold was studied by Margulis, [Mar]. His proof, which is dierent from the above mentioned works, is based on the mixing property of the Margulis measure for the geodesic ow. In the constant negative curvature case, Margulis' method combined with an exponential mixing rate for the geodesic ow, also provides an estimate like (1) | albeit with a weaker power saving, see e.g. [MMO].

1.1.Simple closed geodesics.The aforementioned fundamental results do not provide

any estimates for the number of simple closed geodesics onM. Indeed, very few closed geodesics onMare simple, [BS2], and it is hard to discern them in1(M), [BS1]. More explicitly, it was shown in [Ri] that the number ofsimpleclosed geodesics of length at most

LonMis bounded above and below byOM(L6g6).A.E. acknowledges support by the NSF and the Simons Foundation.

A.M. acknowledges support by the NSF and Alfred P. Sloan Research Fellowship. 1

2 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI

In her PhD thesis, [Mir1] and [Mir2], Mirzakhani proved an asymptotic growth rate for the number of simple closed geodesics of a given topological type on a hyperbolic surfaceM| recall that two simple closed geodesics and

0onMare of the same topological type if

there exists someg2Modgso that 0=g LetXbe a compact surface equipped with a Riemannian metric of negative curvature. We emphasize that the curvature is not assumed to be constant; indeed, elements inM(S) will be denoted byMto minimize the confusion. By a multi-geodesic onXwe mean =Pd i=1ai iwhere i's are disjoint, essential, simple closed geodesics, andai>0 for all

1id. In this case, we dene`X(

) :=Pai`X( ), where`Xdenotes the length function onX. The multi-geodesic will be called integral (resp. rational) ifai2Z(resp.ai2Q).

Given a rational multi-geodesic

0onX, dene

s X(

0;L) := #f

2Modg:

0:`X( )Lg: Mirzakhani, [Mir2, Thm. 1.1], proved the following estimate whenMis a hyperbolic surface: (2)sM( 0;L)n

0(M)L6g6;

wheren

0:M(S)!R+(theMirzakhanifunction) is a continuous proper function; geo-

metric informations carried byn

0are also studied in [Mir2].

In this paper we obtain a quatitative version of (2); moreover, our approach allows us to prove such a result in the more general setting ofvariablenegative curvature. Theorem 1.1.There exists some=(g)>0so that the following holds. LetXbe a compact surface of genusgequipped with a Riemannian metric of negative curvature. Let

0be a rational multi-geodesic onX. Then

s X(

0;L) =n

0(X)L6g6+O

0;X(L6g6)

wheren

0(X)is a positive constant which depends on

0andX.

The proof of Theorem 1.1 is based on the study of a related counting problem in the space of geodesic measured laminations onS, a la Mirzakhani. The space of measured laminations onS, which we denote byML(S), is a piecewise linear integral manifold homeomorphic toR6g6; but it does not have a natural dierentiable structure, [Th1]. Train tracks were introduced by Thurston as a powerful technical device for understanding measured laminations. Roughly speaking, train tracks are induced by squeezing almost parallel strands of a very long simple closed geodesic to simple arcs on a surface; they provide linear charts forML(S).

The mapping class group Mod

gofSacts naturally onML(S). Moreover, there is a natural Mod g-invariant locally nite measure onML(S), the Thurston measureTh, given by the piecewise linear integral structure onML(S), [Th1]. For any open subsetU ML(S) and anyt >0, we have

Th(tU) =t6g6Th(U):

On the other hand, any metric of negative curvatureXonSinduces the length function

7!`X() onML(S), which satises`X(t) =t`X() for allt >0. It is proved in [Mir1,

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 3

App. A] that`Mis a convex function onML(S) whenMis a hyperbolic surface. This fact remains valid in the more general setting of variable negative curvature, seex5.5. The source of the polynomially eective error term in Theorem 1.1 is the exponential mixing property of the Teichmuller geodesic ow proved by Avila, Gouezel, and Yoccoz, [AGY, AR, AG]. We combine this estimate with ideas developed by Margulis in his PhD thesis, [Mar], to prove the following theorem which is of independent interest | see Theorem 7.1 for a more general statement. Letbe a train track and letU() be the corresponding train track chart. For every

2U() we letkkdenote the sum of the weights ofinU(), seex5.

Theorem 1.2.There exists some1=1(g)>0so the following holds. Letbe a maximal train track. LetL1and let

0be a simple close curve onM. There exists a constant

c

0>0so that

#f

2U()\Modg:

0:k kLg=c

0volL6g6+O;

0(L6g61)

wherevol=Thf2U() :kk1g. It is worth noting that in view of Theorem 1.2, the asymptotic behavior of the number of points in one Mod g-orbit in the conef:kkLgand that of the number of integral points in this cone agree up to multiplicative constant. Theorem 1.2, in the more general form Theorem 7.1, plays a crucial role in our analysis. Indeed, using the aforementioned convexity of the length function, we will prove Theorem 1.1 using Theorem 7.1 inx8. It is an intriguing problem to investigate the asymptotic behavior of functions similar to and dierent fromsX(

0;L) or the complexity considered in Theorem 1.2. For instance,

for a suitable formulation of a combinatorial length | using intersection numbers | the count is exactly a polynomial, see [FLP]. We also refer the reader to [CMP] where a related problem is studied for thrice punctured sphere.

1.2.Outline of the paper.Inx2 we collect some preliminary results. Inx3 we prove an

equidistribution result with an error term, Proposition 3.2, which may be of independent interest; see, e.g. [KM, LMir]. The proof of this proposition is based on the exponential mixing rate for the Teichmuller geodesic ow, [AGY], and the so calledthickeningtech- nique, see [Mar, EMc]. Inx4 we prove Proposition 4.1; this proposition is one of the main ingredients in the proof, and could be compared to arguments in [Mar, Chap. 6]. We will recall some basic facts aboutML(S), and study the relation between the linear structures onML(S) and the space of quadratic dierentials inx5 andx6. The orbital counting in sec- tors ofML(S) is studied inx7; the main result here is Theorem 7.1. We prove Theorem 1.1 inx8.

1.3.Acknowledgement.This project originated in fall of 2015 when the authors were

members of the Institute for Advanced Study (IAS), we thank the IAS for its hospitality. We thank C. McMullen, K. Ra, and A. Zorich for helpful discussions. We also thank F. Arana-Herrera, H. Oh, and A. Wright for their comments on an earlier version of this

4 ALEX ESKIN, MARYAM MIRZAKHANI, AND AMIR MOHAMMADI

paper. We are in debt to G. Margulis and F. Arana-Herrera for drawing our attention to the case of variable negative curvature, and to K. Ra for providing the proof of Theorem 5.1. Last, but not least, we thank the anonymous referee for their careful reading and several helpful comments.

2.Preliminaries and notation

LetQ(S) denote the moduli space of quadratic dierentials onS, and letQ1(S) be the

moduli space of quadratic dierentials with area one onS. For any= (1;:::;k;&) withPi= 4g4 and&2 f1g, deneQ1() to be (a connected component) of the stratum

of quadratic dierentials consisting of pairs (M;q) whereM2 M(S) andqis a unit area quadratic dierential onMwhose zeros have multiplicities1;:::;kand&= 1 ifqis the quare of an abelian dierential and1 otherwise. Then Q

1(S) =G

Q 1(): PutQ() :=ftq:t2R;q2 Q1()g. Let Sbe a set ofkdistinct marked points. LetQ1T() denote the space of quadratic dierentials (M;q) equipped with an equivalence class of homeomorphismsf:S!Mthat send the marked points to the zeros ofq. The equivalence relation is isotopy rel marked points. Let:Q1T()! Q1() be the forgetful map which forgets the markingf; this is an innite degree branched covering.

Similarly, let

(S) denote the moduli space of Abelian dierentials onS, and let

1(S) be

the moduli space of area one Abelian dierentials. For any= (1;:::;k), we letH() denote the corresponding stratum, and letH1() denote the area one abelian dierentials.

Note that passing to a branched double cover

^MofM, we may realizeQ1() as anane invariant submanifoldinH1(^) corresponding to odd cohomology classes on^M, seex2.1. However, even ifqbelongs to a compact subset ofQ1(S), the complex structure on^M may have very short closed curves in the hyperbolic metric, e.g. a short saddle connection between two distinct zeros on (M;q) could lift to a short loop in^M. Note however that if ^M;!) is the aforementioned double cover of (M;q), then the length of the shortest saddle connection in!is bounded by the length of the shortest saddle connection inq, i.e. compact subsets ofQ1() lift to compact subsets ofH1(^).

2.1.Period coordinates.Letx= (M;!)2 H(), and let Mbe the set of zeros of

!. Passing to a nite cover, which we continue to denote byH(), we assume there are no orbifold points inH(). Dene the period map :H()!H1(M;;C): Let us recall that can be dened as follows. Let # =k. Fix a triangulationTof the surface by saddle connections ofx, that is: 2g+k1 directed edges1;:::;2g+k1which form a basis forH1(M;;Z). Dene (x) =Z i! 2g+k1 i=1:

EFFECTIVE COUNTING OF SIMPLE CLOSED GEODESICS 5

Note that this map depends on the triangulationT. IfT0is any other triangulation, and

0is the corresponding period map, then 01is linear. For anyx2 H(), there is

a neighborhoodB(x) ofxso that the restriction of toB(x) is a homeomorphism onto (B(x)), seex2.9. We always chooseB(x) small enough so that, using the Gauss-Manin connection, the triangulation aty2B(x) can be identied with the triangulation atx. We dene the period coordinates atx= (M;q)2 Q() as follows. If&= 1, thenqis a square of an abelian dierential, and we may dene period coordinates as above. If&=1, we use the orienting double coverH(^) to dene the period coordinates: in this case there is a canonical injection fromQ() intoH(^). Any Riemann surface in the image of this map is equipped with an involution. This way we get the period map fromQ() toH1odd(M;;C)quotesdbs_dbs33.pdfusesText_39

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