Logic Seminar

For a topological group GG, a GG-flow is a continuous action of GG on a compact Hausdorff space XX; we call XX the phase space of the GG-flow. A GG-flow on XX is minimal if XX has no closed non-trivial invariant subset. The universal minimal GG-flow, M(G)M(G), has every minimal GG-flow as a quotient and it is unique up to isomorphism. We show that whenever we have a short exact sequence0→K→G→H→00→K→G→H→0of topological groups with the image of KK a compact normal subgroup of GG, then the phase space of M(G)M(G) is homeomorphic to the product of the phase space of M(H)M(H) with KK. For instance, if GG is a Polish, non-Archimedean group, and the image of KK is open in GG, then HH is a countable discrete group. The phase space of M(H)M(H) is homeomorphic to Gl(22ℵ0)Gl⁡(22ℵ0), the Stone space of the completion of the free Boolean algebra on 2ℵ02ℵ0 generators by Balcar-Błaszczyk and Glasner-Tsankov-Weiss-Zucker. Therefore, the phase space of M(G)M(G) is homeomorphic to K×Gl(22ℵ0)K×Gl⁡(22ℵ0). When the sequence splits, that is, G≅H⋉KG≅H⋉K, then the homeomorphism witnesses an isomorphism of flows, recovering a result of Kechris and Sokić.