Growth of the number of simple closed geodesics on hyperbolic surfaces. (English) Zbl 1177.37036

Mirzakhani, Maryam

Ann. Math. (2) 168, No. 1, 97-125 (2008).

Let \(X\) be a complete hyperbolic Riemann surface of finite area, genus \(g\), with \(n\) cusps, and let \(\gamma_0\) be a simple closed geodesic on \(X\). Let then \(s(L)\) (respectively, \(s_X(L,\gamma_0)\)) denote the number of simple closed geodesics \(\gamma\) of length \(\leq L\) (respectively, having the same type as \(\gamma_0\), i.e. such that \(X-\gamma\) be homeomorphic to \(X\setminus\gamma_0\)).
We call a multi-curve on \(X\) any formal linear combination \(\gamma:= \sum^k_{i=1} a_i\gamma_i\), with positive rational coefficients \(a_i\), of disjoint, essential, non-peripheral, simple closed curves \(\gamma_i\), which belong to distinct homotopy classes. The length of such \(\gamma\) is defined by \[ \ell_X(\gamma):= \sum^k_{i=1} a_i\ell_X(\gamma_i). \] Then the main result is that, for any multicurve \(\gamma\), \(\lim_{L\to\infty} s_X(L,\gamma)/L^{6g- 6+2n}= N_\gamma(X)\), where \(N_\gamma\) is a positive continuous proper function on the moduli space of complete hyperbolic Riemann surfaces of genus \(g\) with \(n\) cusps. Moreover, the contributions of \(\gamma\) and \(X\) is \(N_\gamma(X)\) separate into: \(N_\gamma(X)= c(\gamma)\times\beta(X)\), where \(c(\gamma)\) is a positive rational depending only on the type \(\gamma\), and \(\beta(X)\) is a normalized measure of the set of multi-curves on \(X\) having integral coefficients and length \(\leq 1\).
Central ingredients of the proof are:
– The Weil-Petersson volume form on the moduli space, and the integral of \(s_X(L,\gamma)\) with respect to it;
– the ergodic action of the mapping class group on the space of measured geodesic laminations (endowed with the Thurston measure), over a closed surface of genus \(g\) with \(n\) boundary components.

Reviewer: Jacques Franchi (Strasbourg)

MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53C22	Geodesics in global differential geometry
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37J50	Action-minimizing orbits and measures (MSC2010)

Keywords:

Riemann surfaces; simple closed geodesics; moduli space; mapping class group; geodesic laminations; Thurston measure; Weil-Peterrson volume form