On free components of Artin and Coxeter groups

TL;DR

This paper investigates the structure of connected components in Artin and Coxeter groups through von Neumann algebras and measure equivalence, highlighting exceptions and specific cases.

Contribution

It provides new insights into the classification of Artin and Coxeter groups using operator algebra techniques and measure theory, addressing known exceptions.

Findings

Connected components are characterized by von Neumann algebras in Artin groups.

For Coxeter groups, results hold without assuming relative hyperbolicity.

Discusses measure equivalence cases involving products of nonabelian free groups.

Abstract

The number of connected components can be remembered by the von Neumann algebra among Artin groups, the only possible exception being the case that corresponds to the free group factor problem. In the case of Coxeter groups, this result is obtained in the absence of relatively hyperbolicity. We also discuss a specific case of the analogous problem in measure equivalence where each factor group is a product of nonabelian free groups.