Strong solidity classification of Coxeter groups

TL;DR

This paper establishes a clear dichotomy for Coxeter groups, showing they either have strongly solid von Neumann algebras or contain specific product groups, with a new streamlined proof and geometric characterizations.

Contribution

It introduces a novel, simplified proof of the dichotomy for Coxeter groups and provides geometric and group-theoretic characterizations for strong solidity.

Findings

Coxeter groups either have strongly solid von Neumann algebras or contain a specific product group.

Complete classification of Coxeter groups with strongly solid von Neumann algebras based on Coxeter-Dynkin diagrams.

Extension of the dichotomy to virtually cocompact special groups.

Abstract

We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled Coxeter groups by Borst-Caspers. However, our proof is conceptually different, which leads to a significantly streamlined argument. We also provide additional equivalent geometric and group-theoretic characterizations of strong solidity for Coxeter groups that allow us to completely classify those with a strongly solid group von Neumann algebra. In particular, we characterize strong solidity purely in terms of the defining Coxeter-Dynkin diagram. Finally, we obtain the same dichotomy for virtually cocompact special groups.