The Geometry of Outer Automorphism Groups of Universal Right-Angled Coxeter Groups

TL;DR

This paper explores the geometric structure of automorphism groups of universal right-angled Coxeter groups, showing that their natural models cannot be endowed with non-positive curvature metrics, thus providing insight into their geometric properties.

Contribution

It demonstrates that the natural combinatorial and topological models for these automorphism groups cannot be equipped with an equivariant CAT(0) metric, revealing limitations in their geometric structure.

Findings

The models are not CAT(0) spaces.

Provides the first non-trivial example of such a space.

Advances understanding of automorphism groups of Coxeter groups.

Abstract

We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer Space as a model for Out(F_n), we prove that the natural combinatorial and topological model for their outer automorphism groups can \emph{not} be given an equivariant CAT(0) metric. This is particularly interesting as there are very few non-trivial examples of proving that a model space of independent interest is not CAT(0).