Simultaneous construction of hyperbolic isometries

TL;DR

This paper establishes conditions under which a single group element acts hyperbolically across multiple hyperbolic spaces and applies this to prove a conjecture in geometric group theory related to automorphisms of free groups.

Contribution

It introduces a sufficient condition for constructing a single hyperbolic isometry compatible with multiple actions and proves a conjecture about automorphisms of free groups.

Findings

Existence of a common hyperbolic element under certain conditions

Resolution of a conjecture by Handel and Mosher

Advancement in understanding automorphisms of free groups

Abstract

Given isometric actions by a group G on finitely many \delta-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for each action. As an application we prove a conjecture of Handel and Mosher regarding relatively fully irreducible subgroups and elements in the outer automorphism group of a free group.