The galaxy of Coxeter groups

TL;DR

This paper introduces the galaxy of Coxeter groups, a new geometric framework for understanding isomorphisms between Coxeter systems, and explores structural properties, classifications, and rigidity results within this space.

Contribution

It proposes a novel geometric complex to study Coxeter group isomorphisms, characterizes small rank cases, and demonstrates profinite rigidity of triangle Coxeter groups.

Findings

Structural results about the galaxy of Coxeter groups

Full classification in small ranks

Profinite rigidity of triangle Coxeter groups

Abstract

In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.