A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

Abstract

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group’s defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group’s order of thickness, order of algebraic thickness and divergence function. Finally, given an integer $n > 1$, we give examples of right-angled Coxeter groups which are thick of order $n$, yet are algebraically thick of order strictly larger than n, answering a question of Behrstock-Druţu-Mosher.