Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents

TL;DR

This paper establishes a uniform bound on exponents for which powers of independent pseudo-Anosov elements generate a free, convex-cocompact subgroup in the mapping class group of a surface.

Contribution

It proves the existence of a universal constant depending on the surface, ensuring that sufficiently large powers of independent pseudo-Anosov elements generate free, convex-cocompact subgroups.

Findings

Existence of a constant M(S) for uniform bounds on exponents.

Subgroups generated by large powers are free and convex-cocompact.

All non-trivial elements in these subgroups are pseudo-Anosov.

Abstract

Let $S$ be a compact orientable surface, and $\Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b \in \Mod(S)$ are independent (i.e., $[a^n,b^m]\not=1$ for any $n,m \not=0$) pseudo-Anosov elements. Then for any $n,m \ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-cocompact if $|n|+|m| \ge N$ and $nm \not=0$.