Ping-pong and Outer space

TL;DR

This paper proves that under certain conditions, high powers of two hyperbolic iwips in Out(F_N) generate a free subgroup where all nontrivial elements are also hyperbolic iwips, analogous to pseudo-Anosov elements in mapping class groups.

Contribution

It establishes that certain pairs of hyperbolic iwips generate free subgroups with all elements being hyperbolic iwips, extending the analogy with pseudo-Anosov elements.

Findings

High powers of hyperbolic iwips generate free subgroups.

All nontrivial elements in these subgroups are hyperbolic iwips.

The result provides an analog of purely pseudo-Anosov free subgroups.

Abstract

We prove that if $\phi,\psi\in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<\phi,\psi>\le Out(F_N)$ is not virtually cyclic then some high powers of $\phi$ and $\psi$ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $Out(F_N)$ is strongly analogous to being a pseudo-Anosov element of a mapping class group, so the above result provides analogs of "purely pseudo-Anosov" free subgroups of $Out(F_N)$.