Counting closed geodesics in moduli space

Eskin, Alex; Mirzakhani, Maryam

doi:10.3934/jmd.2011.5.71

Cited by 47 publications

(74 citation statements)

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“…The flow is proved ergodic and mixing by Masur [Mas82] and Veech [Vee82]. The stratification itself has even been extensively studied, for example in the work of [EMR12], [KZ03], [Lan04], [Lan05], and [Zor10]. Our work particularly relates to the work on determining connected components of a strata, as we show that a given index list naturally divides into many components.…”

Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 94%

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Pfaff

2016

J. Topol. Anal.

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Abstract. By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an Out(Fr) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [HM11] give a strictly finer invariant in the analogous Out(Fr) setting, we determine which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F3). IntroductionLet F r denote the free group of rank r and Out(F r ) its outer automorphism group. In this paper we prove realization results for an invariant dependent only on the conjugacy class (within Out(F r )) of the outer automorphism, namely the "ideal Whitehead graph." 1.1. Main result. A "fully irreducible" (iwip) outer automorphism is the most commonly used analogue to a pseudo-Anosov mapping class and is generic. An element φ ∈ Out(F r ) is fully irreducible if no positive power φ k fixes the conjugacy class of a proper free factor of F r .We give in Subsection 2.3 the exact Out(F r ) definition of an ideal Whitehead graph. For now, to give context, we remark that, for a pseudo-Anosov surface homeomorphism, the component of an ideal Whitehead graph coming from a foliation singularity is a polygon with edges corresponding to the lamination leaf lifts bounding a principal region in the universal cover [NH86].Handel and Mosher define in [HM11] a notion of an ideal Whitehead graph for a fully irreducible outer automorphism, a finite graph whose isomorphism type is an invariant of the conjugacy class of the outer automorphism. In this paper we investigate the extent to which the Out(F r ) situation is more complicated by giving a partial answer to a question posed by Handel and Mosher in [HM11]: Question 1.1. For each r ≥ 2, which isomorphism types of graphs occur as IW(φ) for a fully irreducible φ ∈ Out(F r )? Theorem. A. Exactly eighteen of the twenty-one connected, simplicial five-vertex graphs are the ideal Whitehead graph IW(φ) for a fully irreducible outer automorphism φ ∈ Out(F 3 ).The twenty-one connected, simplicial five-vertex graphs ([CP84]

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Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 94%

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Pfaff

2016

J. Topol. Anal.

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“…The main theme is that the exponential growth rate of such number is equal to the volume entropy, and more precisely the number of primitive closed geodesics of length ≤ t is asymptotically proportional to exp(h vol t) h vol t where t tends to infinity. (See the thesis of G. Margulis [29] for compact negatively curved manifolds, and for more recent results, papers of A. Eskin-M. Mirzakhani [15] for Moduli spaces with Teichmüller flow, E. Makover -J. McGowan [26] for random manifolds, Z. Lian -L.S. Young [24] for mappings of Hilbert spaces, etc.…”

Section: Other Topological Invariant Related To Growthmentioning

confidence: 99%

Entropy Rigidity for Metric Spaces

Lim¹

2012

The Pure and Applied Mathematics

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“…The existence of in Theorem 1.1 is originally due to Veech [35], and it is essentially independent of the rest of the paper. Although the argument has appeared explicitly (see [10,35]), and is implicit in Bers' proof of Thurston's classification of mapping classes [8], we give the proof in Section 9 for completeness. The argument follows the well-known theme in hyperbolic geometry of short geodesics; see McMullen's paper [26] for a broad discussion of this idea.…”

Section: Introductionmentioning

confidence: 99%

“…Veech [35] was the first to study the asymptotic behavior of |G g (L)| as L tends to infinity. His line of work eventually culminated in Eskin-Mirzakhani's asymptotic formula [10], which gives |G g (L)| ∼ e (6g−6)(L/g) (6g − 6)(L/g) ,…”

Section: Introductionmentioning

confidence: 99%

On the number and location of short geodesics in moduli space

Leininger

Margalit

2012

Journal of Topology

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Abstract. A closed Teichmüller geodesic in the moduli space M g of Riemann surfaces of genus g is called L-short if it has length at most L/g. We show that for any L > 0 there exist R > ǫ > 0, independent of g, so that the L-short geodesics in M g all lie in the intersection of the ǫ-thick part and the R-thin part. We also estimate the number of L-short geodesics in M g , bounding this from above and below by polynomials in g whose degrees depend on L and tend to infinity as L does.

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Counting closed geodesics in moduli space

Cited by 47 publications

References 20 publications

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Entropy Rigidity for Metric Spaces

On the number and location of short geodesics in moduli space

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