Division algebras and transitivity of group actions on buildings

Abstract

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× and SL1(D) on the spherical and affine buildings that may be associated to GLd(K) and SLd(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.