  We establish results connecting the uniform Liouville property of group
actions on the classes of a countable Borel equivalence relation with
amenability of this equivalence relation. We also study extensions of Kesten's
theorem to certain classes of topological groups and prove a version of this
theorem for amenable SIN groups. Furthermore, we discuss relationship between
generalizations of Kesten's theorem and anticoncentration inequalities for the
inverted orbits of random walks on the classes of an amenable equivalence
relation. This allows us to construct an amenable Polish group that does not
satisfy the limit conditions in combinatorial extensions of Kesten's theorem.
