Flat rank of automorphism groups of buildings

Abstract

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 CAT(0)-realization |W | 0 . We can prove these inequalities under weaker assumptions. We also show that for any integer n ≥ 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G) = n and which is not linear.

Geometric characterization of flat groups of automorphisms

“…Conjecture 16 in the paper [BRW07] is therefore false as stated; the first and third author of the current paper should have known this while being involved in writing the paper [BRW07], because they obtained Theorem 5.3 below, which also illustrates this point, by different methods in unpublished work done in the year 2000.…”

“…We conclude that the orbits of N and T in B.G .k// are quasi-flats of dimension k -rank.G /. That k -rank.G / D flat-rank.G .k// follows from Corollary 19 in [BRW07].…”

Non-distortion of twin building lattices

“…Note that another notion of rank, relevant to Willis' general theory of totally disconnected locally compact groups, is discussed for the full automorphism groups G ± = Aut(X ± ) in [8], and turns out to coincide with the above notions of rank.…”

Contraction groups in complete Kac–Moody groups

“…The groups covered by the second part of our Main Theorem are topologically simple [15], indeed in many cases algebraically simple [4] groups, whose flat rank assumes all positive integral values [1], and indeed are the first known groups who have non-closed contraction groups and whose flat rank can be larger than 2; we refer the reader to [19,1] for the definition of flat rank. They are thus 'larger' but similar to the group of type-preserving isometries of a regular, locally finite tree, which is a simple, totally disconnected, locally compact group of flat rank 1, whose non-trivial contraction groups are non-closed.…”

“…However, note that the topology and the completion of a group with locally finite twin root datum do not depend on the CAT(0)-structure, only on the combinatorics of the action on the building; see Lemma 2 in [1]. Therefore one should be able to dispense with the use of the Davis-realization below.…”