Compact group actions with the Rokhlin property

Abstract

We provide a systematic and in-depth study of compact group actions with the Rokhlin property. It is show that the Rokhlin property is generic in some cases of interest; the case of totally disconnected groups being the most satisfactory one. One of our main results asserts that the inclusion of the fixed point algebra induces an order-embedding on K-theory, and that it has a splitting whenever it is restricted to finitely generated subgroups. We develop new results in the context of equivariant semiprojectivity to study actions with the Rokhlin property. For example, we characterize when the translation action of a compact group on itself is equivariantly semiprojective. As an application, it is shown that every Rokhlin action of a compact Lie group of dimension at most one is a dual action. Similarly, for an action of a compact Lie group G on C(X), the Rokhlin property is equivalent to freeness together with triviality of the principal G-bundle $X\to X/G$.