In this talk we will discuss several notions of amenability of non-locally compact topological groups and their implications for the properties of countable subgroups of an amenable topological group. In particular, we will prove a combinatorial version of Kesten's theorem for amenable Hausdorff topological groups with small invariant neighborhoods. Moreover, we will describe a family of topological groups which could either provide us with a counterexample to further extensions of Kesten's theorem or bring us closer to proving amenability of the group of the interval exchange transformations. These groups could be viewed as generalizations of lamplighter groups in the measurable setting. If time permits, we will also cover the Liouville property of group actions on the orbits of a countable Borel equivalence relation. The talk is based on a joint work with Kate Juschenko and Martin Schneider.