Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

Abstract

A classical question in the theory of hyperbolic surfaces is the study of lengths of closed geodesics under various constraints. A celebrated result in this area is M. Mirzakhani's asymptotic formula for the number of simple closed geodesics of length $\le L$ on a hyperbolic surface of genus $g$ with $n$ punctures. We investigate the number of simple closed geodesics of length $\le L$ representing a fixed primitive nonzero homology class $x$ on a hyperbolic surface $S$. We denote this number by $h_{S}(L, x)$. It follows from Mirzakhani's result that $h_{S}(L, x) \le C L^{6(g-1) + 2n}$. However, numerical evidence suggests that this bound is apparently not asymptotically sharp. We prove that for a surface $S$ of genus $g$ with $n$ punctures and $b$ geodesic boundary components, under the condition that $g \ge 1$ and $g+n+b \ge 3$, there exists a constant $C_1 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S}(L, x) \ge C_1 L^{6(g-1) + 2(n + b-1)} \] holds. In the special case of a torus with two punctures $S_{1, 2}$, we obtain the following stronger result: there exists a constant $C_2 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S_{1, 2}}(L, x) \ge C_2 L^{3.011057 \ldots } \] holds.