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

Abstract:

Let $S$ be a compact orientable surface, and $\mathrm {Mod}(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b \in \mathrm {Mod}(S)$ 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 $\langle a^n,b^m \rangle$ is a free group freely generated by $a^n$ and $b^m$, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $\langle a^n,b^m \rangle$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $\langle a^n,b^m \rangle$ is a free group freely generated by $a^n$ and $b^m$, and convex-cocompact if $|n|+|m| \ge N$ and $nm \not =0$.