Relative hyperbolicity of free extensions of free groups

TL;DR

This paper establishes criteria for when free-by-free groups are relatively hyperbolic, based on properties of exponentially growing automorphisms and invariant subgroup systems, contributing to understanding their geometric structure.

Contribution

It provides necessary and sufficient conditions for relative hyperbolicity of free extensions of free groups using automorphism dynamics and subgroup systems.

Findings

Conditions for relative hyperbolicity are characterized by exponential growth properties.

Construction of peripheral subgroups relies on invariant subgroup systems and automorphism powers.

Extension groups become hyperbolic under specified automorphism and subgroup conditions.

Abstract

We give necessary and sufficient conditions for a free-by-free group to be relatively hyperbolic with a cusp-preserving structure. Namely, if $\phi_1, \ldots , \phi_k $ is a collection of exponentially growing outer automorphisms with a common invariant \emph{subgroup system} such that any conjugacy class in the complement of this system grows exponentially under iteration by all $\phi_i$, then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of $\mathbb F$ by the free group generated by sufficiently high powers of $\phi_1, \ldots , \phi_k $, will be hyperbolic.