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On geometric flats in the CAT(O) realization of coxeter groups and tits buildings

par Caprace, Pierre-Emmanuel ;Frédéric, Haglund
Référence Canadian journal of mathematics, 61, 4, page (740-761)
Publication Publié, 2009-08

Article révisé par les pairs

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Résumé :

Given a complete CAT(O) space X endowed with a geometric action of a group I\ it is known that if T 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(O) realization of a Coxeter group W, and T 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(O) realization of arbitrary Tits buildings. © Canadian Mathematical Society 2009.