Ping-pong and Outer space

Abstract

We prove that if $\varphi ,\psi \in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<\varphi ,\psi >\le Out(F_N)$ is not virtually cyclic then some high powers of $\varphi$ 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 $Out(F_N)$.

revised version incorporating the referee's suggestions

Cited by in corpus (13)

• Hyperbolic extensions of free groups
• Lines of minima in Outer space
• Algorithmic detectability of iwip automorphisms
• North-South dynamics of hyperbolic free group automorphisms on the space of currents
• Generalized north-south dynamics on the space of geodesic currents
• A note on subfactor projections
• Stabilizers of $\mathbb{R}$-trees with free isometric actions of $F_N$
• Dynamics of hyperbolic iwips
• Subset currents on free groups
• Hyperbolic extensions of free groups from atoroidal ping-pong
• Atoroidal dynamics of subgroups of Out(F_N)
• Uniform independence for Dehn twist automorphisms of a free group
• Patterson-Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for Outer space