On minimal actions of Polish groups

We show the existence of an infinite monothetic Polish topological group G with the fixed point on compacta property. Such a group provides a positive answer to a question of Mitchell who asked whether such groups exist, and a negative answer to a problem of R. Ellis on the isomorphism of L(G), the universal point transitive G-system (for discrete G this is the same as βG the Stone-Čech compactification of G) and E(M,G), the enveloping semigroup of the universal minimal G-system (M,G). For G with the fixed point on compacta property M is trivial while L(G) is not. Our next result is that even for Z with the discrete topology, L(Z) = βZ is not isomorphic to E(M, Z). Finally we show that the existence of a minimally almost periodic monothetic Polish topological group which does not have the fixed point property will provide a negative answer to an old problem in combinatorial number theory.