We apply some of the ideas of Margulis’s Ph.D. dissertation to Teichmüller space. Let be a point in Teichmüller space, and let be the ball of radius centered at (with distances measured in the Teichmüller metric). We obtain asymptotic formulas as tends to infinity for the volume of , and also for the cardinality of the intersection of with an orbit of the mapping class group.
"Lattice point asymptotics and volume growth on Teichmüller space." Duke Math. J. 161 (6) 1055 - 1111, 15 April 2012. https://doi.org/10.1215/00127094-1548443