Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if $\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We prove the 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 $\Gamma$ 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.

Keywords:

Coxeter group, flat rank, $\cat0$ space, building Coxeter group, flat rank, $\cat0$ space, building

MSC Classifications:

20F55, 51F15, 53C23, 20E42, 51E24 show english descriptions Reflection and Coxeter groups [See also 22E40, 51F15]
Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]
Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces
Groups with a $BN$-pair; buildings [See also 51E24]
Buildings and the geometry of diagrams 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]
51F15 - Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]
53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces
20E42 - Groups with a $BN$-pair; buildings [See also 51E24]
51E24 - Buildings and the geometry of diagrams

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