Algebraic Ranks of CAT(0) Groups

TL;DR

This paper investigates the algebraic rank of various CAT(0) groups, providing new insights and proofs, including a novel approach to Coxeter group commensurability, expanding understanding of geometric group theory.

Contribution

It offers new results on the algebraic rank of multiple classes of CAT(0) groups and presents a novel proof regarding Coxeter group commensurability.

Findings

Determined algebraic ranks for right-angled Coxeter and Artin groups

Established properties of groups acting on CAT(0) spaces with isolated flats

Provided a new proof of Coxeter group commensurability

Abstract

We study the algebraic rank of various classes of $\mathrm{CAT}(0)$ groups. They include right-angled Coxeter groups, right-angled Artin groups, relatively hyperbolic groups and groups acting geometrically on $\mathrm{CAT}(0)$ spaces with isolated flats. As one of our corollaries, we obtain a new proof of a result on commensurability of Coxeter groups