Automorphisms of geometric structures associated to Coxeter groups

TL;DR

This paper investigates the automorphism groups of geometric structures related to Coxeter groups, identifying conditions for discreteness and describing their structure as semidirect products, extending prior results.

Contribution

It characterizes when automorphism groups of Coxeter-related structures are discrete and describes their structure, extending previous work by Haglund and Paulin.

Findings

Automorphism groups are discrete for specific Coxeter groups.

Discreet automorphism groups can be expressed as semidirect products.

Extends known results to broader classes of Coxeter groups.

Abstract

In this paper, we consider the automorphism groups of the Cayley graph with respect to the Coxeter generators and the Davis complex of an arbitrary Coxeter group. We determine for which Coxeter groups these automorphism groups are discrete. In the case where they are discrete, we express them as semidirect products of two obvious families of automorphisms. This extends a result of Haglund and Paulin.