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On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
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- Journal:
- Canadian Journal of Mathematics / Volume 61 / Issue 4 / 01 August 2009
- Published online by Cambridge University Press:
- 20 November 2018, pp. 740-761
- Print publication:
- 01 August 2009
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- Article
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Given a complete
$\text{CAT}(0)$ 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 $\text{CAT}(0)$ 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 $\text{CAT}(0)$ realization of arbitrary Tits buildings.
