$\mathrm{CAT}(0)$ Extensions of Right-angled Coxeter Groups

TL;DR

This paper proves that split extensions of right-angled Coxeter groups by finite order automorphisms can act faithfully and geometrically on CAT(0) spaces, extending understanding of their geometric group actions.

Contribution

It demonstrates that such extensions admit faithful, geometric actions on CAT(0) spaces, providing new insights into their geometric structure.

Findings

Split extensions act faithfully on CAT(0) spaces

Automorphisms of finite order induce geometric actions

Extends known actions of Coxeter groups

Abstract

We show that any split extension of a right-angled Coxeter group $W_{\Gamma}$ by a generating automorphism of finite order acts faithfully and geometrically on a $\mathrm{CAT}(0)$ metric space.