Subgroups generated by two pseudo-Anosov elements in a mapping class group. I. Uniform exponential growth

Abstract

Suppose $G$ acts acylindrically by isometries on a $\delta$-hyperbolic graph $\Gamma$. We discuss subgroups generated by two hyperbolic elements in $G$ and give sufficient conditions for them to be free of rank two. We apply our results to the mapping class group $\mathrm{Mod}(S)$ of a compact orientable surface $S$ and its action on the curve graph such that $S$ is non-sporadic. There exists a constant $Q$, depending only on $S$, with the following property. If $a, b \in \mathrm{Mod}(S)$ are pseudo-Anosovs such that $\langle a, b \rangle$ is not virtually cyclic, then there exists $M \gt 0$, which depends on $a, b$, such that either $\langle a^n, b^m \rangle$ is free of rank two for all $n \ge Q, m \ge M$, or $\langle a^m, b^n \rangle$ is free of rank two for all $n \ge Q, m \ge M$ (Theorem 3.1). At the end we ask a question in connection to the uniformly exponential growth of subgroups in a mapping class group (Question 3.4).