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

TL;DR

This paper develops a combinatorial formula for the scale of automorphisms in Weyl-transitive groups acting on buildings and characterizes tidy subgroups, linking algebraic and geometric properties.

Contribution

It introduces a new combinatorial approach to compute the scale and identify tidy subgroups for automorphisms of buildings, connecting group actions with geometric structures.

Findings

Derived a formula for the scale of automorphisms

Established the existence of tidy subgroups as stabilizers of simplices

Characterized simplices with tidy stabilizers via minimal sets and Weyl-distance

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 characterized in 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.