A lattice point problem in Teichmuller Space

Let $T_g$ denote the Teichmuller space of marked compact Riemann surfaces of genus g. Let $\Gamma_g$ be the associated mapping class group. In joint work with Bufetov, Eskin, and Mirzakhani, we calculate the asymptotics of

(i) $\Gamma_g x \cap B(y, R)$, for $x, y \in T_g$, where $B(y, R)$ denotes the ball of radius $R$ in the Teichmuller metric

and

(ii) $vol( B(x, R))$.

The proof is based on the results of Margulis’s thesis.