Compact group actions with the Rokhlin property

TL;DR

This paper systematically studies compact group actions with the Rokhlin property, showing its genericity in certain cases, especially for totally disconnected groups, and explores implications for K-theory and equivariant semiprojectivity.

Contribution

It provides new results on the genericity of the Rokhlin property, K-theory embeddings, and characterizations of actions with the Rokhlin property for compact groups.

Findings

Rokhlin property is generic for totally disconnected groups

Inclusion of fixed point algebra induces order-embedding on K-theory

Rokhlin actions of low-dimensional compact Lie groups are dual actions

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 $X\to X/G$.