We prove that the only finite factor-representations of the HigmanThompson groups {F n,r }, {G n,r } are the regular representations and scalar representations arising from group abelianizations. As a corollary, we obtain that any measure-preserving ergodic action of a simple Higman-Thompson group must be essentially free. Finite factor representations of other classes of groups are also discussed.

The study of spectra of operators of unitary group representations has a long history, remarkable achievements and numerous applications. For instance, the famous Kadison-Kaplanski Conjecture which was proven for the case of amenable groups by Higson and Kasparov in [32] asserts that for a torsion free group G and an element m ∈ C[G] of the group algebra of G the spectrum of λ G (m) is connected, where λ G is the left regular representation of G. The remarkable Kesten's criterion of amenability and the fundamental property (T ) of Kazhdan can be formulated in terms of spectral properties of operators of the form λ G (m). The topic in discussion is related to the spectral theory of graphs and networks, random walks, theory of operator algebras, discrete potential theory, abstract harmonic analysis etc.There are three important types of unitary representations associated to a measure class preserving action of a countable group G on a probability space (X, µ): quasi-regular, Koopman and groupoid representations. The goal of this article is to show that there is a close relation between spectral properties of these three types of representations.

In the paper we completely describe characters (central positive-definite functions) of simple locally finite groups that can be represented as inductive limits of (products of) symmetric groups under block diagonal embeddings. Each such group G defines an infinite graded graph that encodes the embedding scheme. The group G acts on the space X of infinite paths of the associated graph by changing initial edges of paths. Assuming the finiteness of the set of ergodic measures for the system (X, G), we establish that each non-regular indecomposable character χ : G → C is uniquely determined by the formula χ(g) = μ 1 (Fix(g)) α 1 · · · μ k (Fix(g)) α k , where μ 1 , . . . , μ k are G-ergodic measures, Fix(g) = {x ∈ X: gx = x}, and α 1 , . . . , α k ∈ {0, 1, . . .}. We illustrate our results on the group of rational permutations of the unit interval. Published by Elsevier Inc.

A. We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than 2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.