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arXiv: math.GR/0510290

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Flat rank of automorphism groups of buildings

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

(2006-01-26)

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.

1:	School of Mathematical and Physical Sciences
	The University of Newcastle
2:	Institut Camille Jordan (ICJ)
	CNRS : UMR5208 – Université Claude Bernard - Lyon I – Ecole Centrale de Lyon – Institut National des Sciences Appliquées (INSA) - Lyon

Subject

:

Mathematics/Group Theory

Keyword(s): totally disconnected group – flat rank – automorphism group – scale function – twin building – strong transitivity – Kac-Moody group

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From: Bertrand Remy <>

Submitted on: Thursday, 26 January 2006 09:55:43

Updated on: Wednesday, 11 April 2007 16:32:44