On geometric flats in the CAT(0) realization of Coxeter groups and Tits buildings

الأصل · EN

Given a complete CAT(0) space X endowed with a geometric action of a group Γ, it is known that if Γ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove a converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Γ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.