Lattice point asymptotics and volume growth on Teichmüller space

Abstract

We apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space. Let $X$ be a point in Teichmüller space, and let $B_R\left(X\right)$ be the ball of radius $R$ centered at $X$ (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as $R$ tends to infinity for the volume of $B_R\left(X\right)$, and also for the cardinality of the intersection of $B_R\left(X\right)$ with an orbit of the mapping class group.