Tree languages and branched groups

TL;DR

This paper explores the relationship between tree isometries, language theory, and self-similar groups, providing characterizations and decidability results for problems involving branched groups and their actions on rooted trees.

Contribution

It introduces a novel characterization of regularly branched self-similar groups using ω-regular languages and establishes the decidability of key problems in this context.

Findings

Characterization of regularly branched self-similar groups via ω-regular languages

Decidability of group comparison and orbit structure problems

Algorithmic methods for analyzing group actions on rooted trees

Abstract

We study the portraits of isometries of rooted trees - the labelling of the tree, at each vertex, by the permutation of its descendants - in terms of languages. We characterize regularly branched self-similar groups in terms of $\omega$-regular languages. We deduce the algorithmic decidability of some problems, such as the comparison of regularly branched contracting groups, and their orbit structure on the boundary of the rooted tree.