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VOL. 52 | 2008 Subgroups generated by two pseudo-Anosov elements in a mapping class group. I. Uniform exponential growth
Koji Fujiwara

Editor(s) Robert Penner, Dieter Kotschick, Takashi Tsuboi, Nariya Kawazumi, Teruaki Kitano, Yoshihiko Mitsumatsu

## 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).

## Information

Published: 1 January 2008
First available in Project Euclid: 28 November 2018

zbMATH: 1170.57017
MathSciNet: MR2509713

Digital Object Identifier: 10.2969/aspm/05210283