Counting finite-order lattice points in Teichmüller space

I will discuss a counting problem for the orbit of the mapping class group in Teichmüller space. Athreya, Bufetov, Eskin, and Mirzakhani have shown that the number of orbit points in a Teichmüller ball of radius R grows like e^{hR}, where h is the dimension of Teichmüller space. Maher has shown that pseudo-Anosov mapping classes are "generic" in the sense that the proportion of these points that are translates by pseudo-Anosovs tends to 1 as R tends to infinity. We aim to quantify this genericity by showing that the number of translates by finite-order and reducible elements have strictly smaller exponential growth rate. In particular, we find that the number of finite-order orbit points grows like e^{hR/2}. Joint work with Howard Masur.

 

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