Topological rigidity in totally disconnected locally compact groups

TL;DR

This paper extends topological rigidity results from semisimple Lie groups to groups of rational points of algebraic groups over non-archimedean local fields, using two novel methods involving group properties and automorphisms of buildings.

Contribution

It introduces two methods to prove topological rigidity for these groups, broadening the class of groups known to have unique compatible topologies.

Findings

Topological rigidity holds for rational points of absolutely quasi-simple algebraic groups over non-archimedean local fields.

The first method applies to groups with specific algebraic and topological properties, ensuring rigidity.

The second method proves rigidity for automorphism groups of regular locally finite buildings.

Abstract

In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $\sigma$-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, $\sigma$-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.