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

摘要

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 characterized in terms of the minimal set of the isometry induced by the element on the Davis realization 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. (C) 2018 Elsevier Inc. All rights reserved.