Scale and tidy subgroups for Weyl-transitive automorphism groups of buildings

U. Baumgartner and J. Parkinson and J. Ramagge

Abstract

We consider closed, Weyl-transitive groups of automorphisms of thick buildings. For each element of such a group, we derive a combinatorial formula for its scale and establish the existence of a tidy subgroup for it that equals the stabilizer of a simplex. Simplices whose stabilizers are tidy for some element of the group are characterizedin terms of the minimal set of the isometry induced by the element on the Davis-realisation of the building and in terms of the Weyl-distance between them and their image. We use our results to derive some topological properties of closed, Weyl-transitive groups of automorphisms.

Keywords: Building, scale, tidy subgroup.

AMS Subject Classification: Primary 22D05; secondary 52E24, 20E36, 20F65.