Non-abelian extensions of minimal rotations

TL;DR

This paper investigates continuous extensions of minimal rotations on compact groups by Lie groups, establishing regularity in specific cases and discussing an open problem related to cocycles on homogeneous spaces.

Contribution

It extends the understanding of regularity for non-abelian extensions of minimal rotations beyond nilpotent groups, highlighting key open questions.

Findings

Proves regularity for certain non-nilpotent Lie group extensions

Identifies conditions under which orbit closures project onto the entire base

Discusses an open problem on cocycles on homogeneous spaces

Abstract

We consider continuous extensions of minimal rotations on a locally connected compact group X by arbitrary locally compact Lie groups and prove regularity (i.e. the existence of orbit closures which project onto the whole basis X) in certain special situations beyond the already known nilpotent case. We further discuss an open question on cocycles acting on homogeneous spaces which seems to be the missing key for a general regularity theorem.