On the non-discreteness of automorphism groups of Cayley graphs of Coxeter groups

TL;DR

This paper characterizes when Cayley graphs of Coxeter groups have uncountable automorphism groups, identifying specific groups with non-discrete automorphism groups and constructing examples of infinite degree vertex-transitive graphs with locally compact automorphism groups.

Contribution

It provides a complete characterization of Coxeter groups whose Cayley graphs have non-discrete automorphism groups and constructs new examples of infinite degree vertex-transitive graphs.

Findings

Identifies Coxeter groups with non-discrete automorphism groups of Cayley graphs

Fully characterizes finitely generated Coxeter groups with non-discrete automorphism groups

Constructs explicit examples of infinite degree vertex-transitive graphs with locally compact automorphism groups

Abstract

In this work we characterise Cayley graphs of Coxeter groups with respect to the standard generating set that admit uncountable vertex stabilisers. As a corollary, we fully identify finitely generated Coxeter groups for which the automorphism group of their Cayley graph with respect to the standard generating set is not discrete when equipped with the permutation topology. As an application, we also provide new explicit constructions of vertex-transitive graphs of infinite degree that have locally compact automorphism groups.