2012/10/04 by Rupert McCallum, McCallum, Rupert
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.1210.1405
In \citeKramer11 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.