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

2023/12/20 by Artem Dudko, Dudko, Artem, Rostislav Grigorchuk +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Representation Theory (math.RT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2312.13124

openalex publication_date 2023/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the construction by Bencs and Tóth 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.

Related

• Random subgroups of branch groups
• On diagonal actions of branch groups and the corresponding characters
• Invariant random subgroups of groups acting on rooted trees
• A family of fractal non-contracting weakly regular branch groups
• Characters and invariant random subgroups of the finitary symmetric group