Algebraic ranks of $\mathrm{CAT}(0)$ groups

Abstract

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