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

Let $S$ be a compact orientable surface, and $\text{Mod}(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b\in \text{Mod}(S)$ are independent (i.e., $[a^n,b^m]≠1$ for any $n,m≠0$) 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 $\mathrm{\mid }n\mathrm{\mid }+\mathrm{\mid }m\mathrm{\mid }\ge N$ and $nm≠0$.