Flat rank of automorphism groups of buildings

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.