Strong solidity classification of Coxeter groups

TL;DR

The authors classify when the group von Neumann algebra of a Coxeter group is strongly solid by proving a dichotomy: absence of a $\mathbb{Z} \times F_2$ subgroup is equivalent to strong solidity, biexactness, and relative hyperbolicity relative to amenable or virtually abelian subgroups. Their approach uses geometric and analytic properties rather than von Neumann algebra techniques, yielding a streamlined proof and diagrammatic criteria from the Coxeter–Dynkin diagram. They extend the dichotomy to virtually cocompact special groups and connect the criteria to established results on relative hyperbolicity and biexactness, providing a complete classification in this broad setting.

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.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• proof : Proof of \ref{['thm:main']}