Counting mapping classes by Nielsen-Thurston type

This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichm\"uller space with the Teichm\"uller metric. In that problem the goal is to count the number of mapping classes that send a given point $x$ into the ball of radius $R$ centered about another point $y$. For the action of the entire group, Athreya, Bufetov, Eskin and Mirzakhani have shown this quantity is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichm\"uller space. We refine the problem by considering the action various distinguished subsets of elements and counting these separately. For the set of finite-order elements, we show the associated count grows coarsely at the rate of $e^{hR/2}$, that is, with half the exponent. For the reducible elements, the associated count grows coarsely at the rate of $e^{(h-1)R}$. Finally, for the set of all multitwists, the coarse growth rate is also $e^{hR/2}$. To obtain these quantitative estimates, we introduce a new notion in Teichm\"uller geometry, called complexity length, which reflects some aspects of the negative curvature of curve complexes and also has applications to counting problems.