We will talk about relations between ergodic properties of group actions and the structure of group characters. The latter is equivalent to the classification of all finite-type factor representations. The outstanding conjecture (often attributed to Vershik) is that for a large class of groups their group characters must have the form \mu(Fix(g)), where $\mu$ is a G-invariant measure for some special group action on a measure space, Fix(g) is the set of all fixed points of group element $g$. We will then establish Vershik's conjecture for the family of Higman-Thompson groups. Since these groups have no non-trivial ergodic measures, we get that they have no non-trivial factor representations. Examples of other classes of groups known to satisfy Vershik's conjecture will be also discussed. The talk will be based on two recent preprints by Dudko and Medynets, "Finite factor representations of Higman-Thompson Groups" ArXiv 1212.1230 and "On characters of inductive limits of symmetric groups" Arxiv 1105.6325