Counting closed geodesics in strata

Abstract

We compute the asymptotic growth rate of the number N (C, R) of closed geodesics of length ≤ R in a connected component C of a stratum of quadratic differentials. We prove that, for any 0 ≤ θ ≤ 1, the number of closed geodesics γ of length at most R such that γ spends at least θ-fraction of its time outside of a compact subset of C is exponentially smaller than N (C, R). The theorem follows from a lattice counting statement. For points x, y in the moduli space M(S) of Riemann surfaces, and for 0 ≤ θ ≤ 1 we find an upper-bound for the number of geodesic paths of length ≤ R in C which connect a point near x to a point near y and spend at least a θ-fraction of the time outside of a compact subset of C.

The Farrell–Jones Conjecture for mapping class groups

“…The question when Teichmüller rays are parallel, or asymptotic, is well understood. See [26,44,52,58,68].…”

Statistically Convex-Cocompact Actions of Groups with Contracting Elements

“…(1) Minsky's result that a pseudo-Anosov element is contracting [57]; (2) the fact that the group action of mapping class groups on Teichmüller space is SCC, which follows from a deep theorem of Eskin, Mirzakhani and Rafi [33,Theorem 1.7], as observed in [2,Section 10].…”

Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation?

“…At our best knowledge, the following theorem is independently due to Eskin-Mirzakhani-Rafi [5] and Hamenstädt [9] (which is rephrased for our purpose).…”

“…We give a brief explanation of the above statement. In [5] and [9], it was stated that when restricted to each connected component of the strata of the moduli space of area one abelian differentials on S n ,…”

“…Here ℓ is the maximum number of zeros of an area one abelian differential on S n , so it is 2n − 2. See [16], [22], [4] and [5] for the relevant background of this theorem.…”