On geometric flats in the {${\rm CAT}(0)$} realization of Coxeter groups and Tits buildings

Details On geometric flats in the {${\rm CAT}(0)$} realization of Coxeter groups and Tits buildings On geometric flats in the {${\rm CAT}(0)$} realization of Coxeter groups and Tits buildings

2006-07-28

arxiv.org/abs/math/0607741v1

Canadian Journal of Mathematics (Journal Canadien de Math\'ematiques) 61, 4 (2009) 740-761

Mathematics

Group Theory

19 pages

Scientific paper

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 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 $\Gamma$ 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.

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