Nilpotent extensions of minimal homeomorphisms

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well known topological essential range if the group is abelian. We define then a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, supposed that their ranges are in a nilpotent group, and then their essential ranges are almost connected.