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# On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

Published:2009-08-01
Printed: Aug 2009
• Pierre-Emmanuel Caprace
• Frédéric Haglund
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## Abstract

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
 MSC Classifications: 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