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On geometric flats in CAT(0) realization of Coxeter groups and Tits buildings

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Caprace, Pierre-Emmanuel [UCL] Haglund, Frédéric [Orsay]

Given a complete CAT(0) space endowed with a geometric action of a group , it is known that if contains a free abelian group of rank , then contains a geometric flat of dimension . We prove the converse of this statement in the special case where is a convex subcomplex of the CAT(0) realization of a Coxeter group , and is a subgroup of . 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.

Document type Article de périodique (Journal article)
Publication date 2009
Language Anglais
Journal information "Canadian Journal of Mathematics" - Vol. 61, no. 4, p. 740-761 (2009)
Peer reviewed yes
Publisher University of Toronto Press * Journals Division ((Canada) Toronto)
issn 0008-414X
e-issn 1496-4279
Publication status Publié
Affiliations UCL - SST/IRMP - Institut de recherche en mathématique et physique
Orsay - Département de mathématiques
Links
Bibliographic reference Caprace, Pierre-Emmanuel ; Haglund, Frédéric. On geometric flats in CAT(0) realization of Coxeter groups and Tits buildings. In: Canadian Journal of Mathematics, Vol. 61, no. 4, p. 740-761 (2009)
Permanent URL http://hdl.handle.net/2078.1/94544