Flat rank of automorphism groups of buildings

Udo Baumgartner (1) , Bertrand Rémy (2) , George A. Willis (1)

1 School of Mathematical and Physical Sciences
2 ICJ - Institut Camille Jordan

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities: alg-rk(W)\le flat-rk(G)\le rk(|W|_0). Here, alg-rk(W) is the maximal $\mathbb{Z}$-rank of abelian subgroups of W, and rk(|W|_0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|_0. We can prove these inequalities under weaker assumptions. We also show that for any integer n \geq 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G)=n and which is not linear.

Mots clés: totally disconnected group; flat rank; automorphism group; scale function; twin building; strong transitivity; Kac-Moody group

Domaines: Théorie des groupes [math.GR]

Udo Baumgartner, Bertrand Rémy, George A. Willis. Flat rank of automorphism groups of buildings. 2006. ⟨hal-00017880⟩