Transformation Groups

, Volume 12, Issue 3, pp 413–436 | Cite as

Flat rank of automorphism groups of buildings

  • Udo BaumgartnerEmail author
  • Bertrand RemyEmail author
  • George A. WillisEmail author


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) ≤ flat-rk(G) ≤ rk(|W|0). Here, alg-rk(W) is the maximal 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 ≥ 1 there is a simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.


Compact Group Weyl Group Coxeter Group Open Subgroup Compact Open Subgroup 
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Copyright information

© Birkhauser Boston 2007

Authors and Affiliations

  1. 1.School of Mathematical and Physical Sciences, The University of Newcastle, University Drive, Building VCallaghan, NSW 2308Australia
  2. 2.Universite de Lyon, Lyon, F-69003, Universite de Lyon 1, Institut Camille Jordan, F-69622 and CNRS, UMR 5208Villeurbanne, F-69622France

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