Geodesic
On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings
Canadian journal of mathematics, Tome 61 (2009) no. 4, pp. 740-761

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2009-040-8
Mots-clés : 20F55, 51F15, 53C23, 20E42, 51E24, Coxeter group, flat rank, CAT(0) space, building 
Caprace, Pierre-Emmanuel; Haglund, Frédéric. On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings. Canadian journal of mathematics, Tome 61 (2009) no. 4, pp. 740-761. doi: 10.4153/CJM-2009-040-8
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