Buildings with isolated subspaces and relatively hyperbolic Coxeter groups

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.

An erratum, correcting two errors in the original paper, has been appended

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