Topological rigidity in totally disconnected locally compact groups

Abstract

In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $σ$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising this to the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field (the second method only achieves this on the additional hypothesis that the group is isotropic). The first method of argument involves demonstrating that, given any topological group $G$ which is totally disconnected, locally compact, $σ$-compact, locally topologically finitely generated, and has the property that no compact open subgroup has an infinite abelian continuous quotient, the group $G$ is topologically rigid in the previously described sense. Then the desired conclusion for the group of rational points of an absolutely quasi-simple algebraic group over a non-archimedean local field may be inferred as a special case. The other method of argument involves proving that any group of automorphisms of a regular locally finite building, which is closed in the compact-open topology and acts Weyl transitively on the building, has the topological rigidity property in question. This again yields the desired result in the case that the group is isotropic.