Pseudo-Anosov stretch factors and homology of mapping tori

TL;DR

This paper investigates pseudo-Anosov elements of the mapping class group that fix a subgroup of the surface's homology, establishing bounds on their entropy and growth rates of conjugacy classes, thus extending previous results.

Contribution

It provides bounds on minimal entropy for pseudo-Anosov elements fixing a homology subgroup and characterizes the growth of conjugacy classes as a polynomial in genus g.

Findings

Smallest entropy is comparable to (k+1)/g.

Number of conjugacy classes grows polynomially with degree k.

Interpolates between known cases for k=0 and k=2g.

Abstract

We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k=0 and k=2g, respectively, and answers a question of Ellenberg. We also show that the number of conjugacy classes of pseudo-Anosov mapping classes as above grows (as a function of g) like a polynomial of degree k.