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 \[ h_{S_{1, 2}}(L, x) \ge C_2 L^{3.011057 \ldots } \] holds.