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

TL;DR

This paper investigates Property (NL) in Coxeter groups, showing that right-angled Coxeter groups have this property only if their defining graph is complete, and classifies triangle groups with Property (NL).

Contribution

It provides a complete characterization of Property (NL) in right-angled Coxeter groups and extends the classification to certain triangle groups.

Findings

RACGs have Property (NL) iff their defining graph is complete.

Disconnected graphs lead to Coxeter groups without Property (NL).

Classification of triangle groups with Property (NL).

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 $\mathrm{(NL)}$.