Characters and IRS's on branch groups and embeddings into hyperfinite factor

Abstract

Using the construction by Bencs and T\'{o}th of invariant random subgroups on weakly branch groups acting on regular rooted trees we produce uncountably many indecomposable characters on these groups. In fact, we study three types of characters coming from the action of a weakly branch group on a regular tree, paying attention to their similarities and differences. We use obtained results to show that each countable amenable branch group has uncountably many pairwise not quasi-equivalent embeddings into Murray-von Neumann hyperfinite factor. For the canonical character associated with a self-similar group and studied by the second author as a self-similar trace we provide a number of examples when it is explicitly computed.