Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

TL;DR

This paper characterizes when Coxeter groups are relatively hyperbolic based on their diagrams, linking this to the isolation of flats in Davis complexes and sub-buildings, with implications for geometric group theory.

Contribution

It provides necessary and sufficient diagram conditions for Coxeter groups to be relatively hyperbolic with respect to parabolic subgroups.

Findings

Coxeter groups are relatively hyperbolic under specific diagram conditions.

Maximal flats in Davis complexes can be isolated based on these conditions.

Isolation of flats extends to sub-buildings of type (W,S).

Abstract

Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of $(W,S)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $(W,S)$ are isolated.