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

Pierre-Emmanuel Caprace (1) , Frédéric Haglund (2)

1 ULB - Département de Mathématique [Bruxelles]
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Coxeter group; flat; flat rank; ${\rm CAT}(0)$ space; building

Domaines: Théorie des groupes [math.GR]

Pierre-Emmanuel Caprace, Frédéric Haglund. On geometric flats in the {${\rm CAT}(0)$} realization of Coxeter groups and Tits buildings. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, 2009, 61 (4), pp.740-761. ⟨hal-00088001⟩