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

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 $$h_{S_{1,2}}(L,x)\ge C_2{L}^{3.011057\dots }$$ holds.