Counting Mapping Class group orbits on hyperbolic surfaces
By Maryam Mirzakhani. 2016
71 citations recorded by the source corpus.
Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\gamma]$ of a closed curve $\gamma \in \pi_{1}(S_{g,n})$ contains a unique closed geodesic on $X$. Let $\ell_{\gamma}(X)$ denote the hyperbolic length of the geodesic representative of $\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length $\leq L$ on $X$ with at most $k$ self-intersections. We also discuss properties of random pants decomposition of large length on $X$. Both these results are based on ergodic properties of the earthquake flow on a natural bundle over the moduli space $\mathcal{M}_{g,n}$ of hyperbolic surfaces of genus $g$ with $n$ cusps.
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Citations in the corpus
- Counting curves in hyperbolic surfaces (2015, methodology): “( 2;) 1g) for every multicurve 0 in . Here Thu is as always the Thurston measure on the space of measured laminations ML() . Remark. In view of Corollary 4.4, it follows from Mirzakhani’s result [14] on the existence of limit (1.1) that the limit (1.3) also exists for any possible lling current for”