Division algebras and transitivity of group actions on buildings

TL;DR

This paper investigates the actions of division algebra groups on buildings, showing they are never strongly transitive for degrees greater than two, but can be Weyl transitive in affine cases, extending previous results.

Contribution

It extends the understanding of group actions on buildings for division algebras of degree greater than two, providing explicit constructions and demonstrating the failure of strong transitivity.

Findings

Actions are never strongly transitive for d>2

Examples of Weyl transitivity in affine cases

Explicit constructions illustrating the failure of strong transitivity

Abstract

Let D be a division algebra with center F and degree d>2. Let K|F be any splitting field. We analyze the action of D^\times and SL_1(D) on the spherical and affine buildings that may be associated to GL_d(K) and SL_d(K), and in particular show it is never strongly transitive. In the affine case we find examples where the action is nonetheless Weyl transitive. This extends results of Abramenko and Brown concerning the d=2 case, where strong transitivity is in fact possible. Our approach produces some explicit constructions, and we find that for d>2 the failure of the action to be strongly transitive is quite dramatic.