On tameness of almost automorphic dynamical systems for general groups

Let $(X,G)$ be a minimal equicontinuous dynamical system, where $X$ is a compact metric space and $G$ some topological group acting on $X$. Under very mild assumptions, we show that the class of regular almost automorphic extensions of $(X,G)$ contains examples of tame but non-null systems as well as non-tame ones. To do that, we first study the representation of almost automorphic systems by means of semicocycles for general groups. Based on this representation, we obtain examples of the above kind in well-studied families of group actions. These include Toeplitz flows over $G$-odometers where $G$ is countable and residually finite as well as symbolic extensions of irrational rotations.