Lifting recursion properties through group homomorphisms

Abstract:

The problem of lifting recursion properties is studied for the case of homomorphisms between transformation groups with different acting groups. It is shown that if the spaces are the same and the group homomorphism is compact-covering then almost periodicity and recurrence lift. This result is applied to certain group actions on covering spaces and on fibre bundles with totally disconnected bundle groups induced by group actions on the base spaces. In particular, it is shown that under suitable conditions if a point in the base space of a fibre bundle with finite bundle group is almost periodic recurrent under the action of a group $G$ then every point in the fibre over the given point is almost periodic recurrent under the induced action of a covering group $\tilde {G}$.