Universal minimal flows of extensions of and by compact groups.

Every topological group G has, up to isomorphism, a unique minimal G -flow that maps onto every minimal G -flow, the universal minimal flow $M(G).$ We show that if G has a compact normal subgroup K that acts freely on $M(G)$ and there exists a uniformly continuous cross-section from $G/K$ to $G,$ then the phase space of $M(G)$ is homeomorphic to the product of the phase space of $M(G/K)$ with K. Moreover, if either the left and right uniformities on G coincide or G is isomorphic to a semidirect product $G/K\ltimes K$ , we also recover the action, in the latter case extending a result of Kechris and Sokić. As an application, we show that the phase space of $M(G)$ for any totally disconnected locally compact Polish group G with a normal open compact subgroup is homeomorphic to a finite set, the Cantor set $2^{\mathbb {N}}$ , $M(\mathbb {Z})$ , or $M(\mathbb {Z})\times 2^{\mathbb {N}}.$ [ABSTRACT FROM AUTHOR]