Geometry of extensions of free groups via automorphisms with fixed points on the complex of free factors

TL;DR

This paper investigates the geometric properties of extensions of free groups under automorphisms, establishing conditions for hyperbolicity using dynamics on the complex of free factors and fixed point analysis.

Contribution

It introduces new criteria for hyperbolicity of free group extensions by analyzing automorphisms with fixed points on the free factor complex.

Findings

Conditions for hyperbolicity and relative hyperbolicity of free group extensions.

Analysis of automorphisms with fixed points on the free factor complex.

Extension geometry when fixed points are sufficiently far apart.

Abstract

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of $\out$ on the complex of free factors combined with the weak attraction theory. We work with subgroups of exponentially growing outer automorphisms and instead of using a standard pingpong argument with loxodromics, we allow fixed points for the action and investigate the geometry of the extension group when the fixed points of the automorphisms on the complex of free factors are sufficiently far apart.