  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.
