Acylindrical Hyperbolicity of Out($W_n$)

TL;DR

This paper proves that the outer automorphism group of the free Coxeter group $W_n$ is acylindrically hyperbolic, revealing geometric properties and implications for CAT(0) spaces with such group actions.

Contribution

It establishes the acylindrical hyperbolicity of Out($W_n$), extending known results from free groups to Coxeter groups and exploring geometric consequences.

Findings

Out($W_n$) is acylindrically hyperbolic

Any CAT(0) space with Out($W_n$) action contains a rank-one geodesic

Links to automorphism groups of free groups

Abstract

We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. As an application, we observe that any CAT(0) space admitting a geometric action by Out($W_n$) must contain a rank-one geodesic. The theorem proceeds from expanding on a well-known relationship between Out($W_n$) and the outer automorphism group of free groups.