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1. CJM 2009 (vol 61 pp. 740)

Caprace, Pierre-Emmanuel; Haglund, Frédéric
On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
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
Categories:20F55, 51F15, 53C23, 20E42, 51E24

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