Property $(\mathrm{NL})$ in Coexeter groups

Abstract

A group has Property $(\mathrm{NL})$ if it does not admit a loxodromic element in any hyperbolic action. In other words, a group with this property is inaccessible for study from the perspective of hyperbolic actions. This property was introduced by Balasubramanya, Fournier-Facio and Genevois, who initiated the study of this property. We expand on this research by studying Property $(\mathrm{NL})$ in Coxeter groups, a class of groups that are defined by an underlying graph. One of our main results show that a right-angled Coxeter group (RACG) has Property $(\mathrm{NL})$ if and only if its defining graph is complete. We then move beyond the right-angled case to show that if a defining graph is disconnected, its corresponding Coxeter group does not have Property $(\mathrm{NL})$. Lastly, we classify which triangle groups (Coxeter groups with three generators) have Property $(\mathrm{NL})$.

11 pages, 6 figures