Lattice Point Asymptotics and Volume Growth on Teichmuller space
  We apply some of the ideas of the Ph.D. Thesis of G. A. Margulis to Teichmuller space. Let x be a point in Teichmuller space, and let B_R(x) be the ball of radius R centered at x (with distances measured in the Teichmuller metric).   We obtain asymptotic formulas as R tends to infinity for the volume of B_R(x), and also for for the cardinality of the intersection of B_R(x) with an orbit of the mapping class group. 
Jayadev Athreya, Alexander Bufetov, Alex Eskin and Maryam Mirzakhani