Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

Abstract

Let $\left(W,S\right)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $\left(W,S\right)$ 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 $\left(W,S\right)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $\left(W,S\right)$ are isolated.