Lattice point asymptotics and volume growth on Teichmüller space (Q411724)

Let \(S_g\) be a closed topological surface of genus \(g \geq 2\) and denote by \(\mathcal{T}_g\) and \(\mathcal{M}_g\) the Teichmüller and moduli spaces of genus \(g\) surfaces, respectively. Let \(B_R(X)\) denote a ball of radius \(R>0\) centred at \(X \in \mathcal{T}_g\). Using ideas of Margulis, in the paper under review the authors find asymptotic formulas for the volume of \(B_R(X)\) as \(R \to \infty\) and also of the cardinality of the intersection of \(B_R(X)\) with the orbit of an element of the mapping class group. In particular, it is shown that NEWLINE\[NEWLINE\begin{aligned} | \Gamma . Y \cap B_R(X) | &\sim \frac{1}{(6g-6)m(\mathcal{M}_g)} \cdot \Lambda(X) \Lambda(Y) e^{(6g-6)R} \quad\text{and}\\ m(B_R(X)) &\sim \frac{1}{(6g-6)m(\mathcal{M}_g)} \cdot e^{(6g-6)R} \Lambda(X) \int_{\mathcal{M}_g} \Lambda (Y) dm (Y)\end{aligned} NEWLINE\]NEWLINE as \(R \to \infty\). Here, \(\Gamma\) is the mapping class group of \(S_g\), \(\Lambda\) is the Hubbard-Masur function and \(6g-6\) is the entropy of the Teichmüller geodesic flow with respect to Lebesgue measure \(m\).