Growth rate of Dehn twist lattice points in Teichmuller space

Athreya et al. [Lattice point asymptotics and volume growth on Teichmuller space, Duke Math. J. 161 (2012) 1055-1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius R in Teichmuller space is asymptotic to e(hR), where h is the dimension of the Teichmuller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius R is coarsely asymptotic to e(h/2 R). Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius R grows coarsely at least at the rate of R & sdot;e(h/2 R).