Piecewise strongly proximal actions, free boundaries and the Neretin groups

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure conjugacy class in Chabauty space does not contain trivial subgroup. We establish dynamical criterion on action totally disconnected $X$ ensuring that no relatively amenable can be confined. This property equivalent to fact its Furstenberg boundary free. Our applies Neretin groups. deduce each has two inequivalent irreducible unitary representations are weakly equivalent. implies groups type I, thereby answering question Y.~Neretin.