Bipolar Coxeter groups

TL;DR

This paper introduces bipolar Coxeter groups, a class characterized by a specific geometric property of their Cayley graphs, and provides a unique conjugacy class result and diagrammatic characterization.

Contribution

It defines bipolar Coxeter groups, proves their unique Coxeter generating set conjugacy class, and characterizes them via their Coxeter diagrams.

Findings

Bipolar Coxeter groups include virtually Poincare duality and infinite irreducible 2-spherical groups.

They have a unique conjugacy class of Coxeter generating sets.

Characterization of bipolar Coxeter groups through Coxeter diagrams.

Abstract

We consider the class of those Coxeter groups for which removing from the Cayley graph any tubular neighbourhood of any wall leaves exactly two connected components. We call these Coxeter groups bipolar. They include both the virtually Poincare duality Coxeter groups and the infinite irreducible 2-spherical ones. We show in a geometric way that a bipolar Coxeter group admits a unique conjugacy class of Coxeter generating sets. Moreover, we provide a characterisation of bipolar Coxeter groups in terms of the associated Coxeter diagram.