A Language Hierarchy and Kitchens-Type Theorem for Self-Similar Groups

TL;DR

This paper extends the concept of self-similar groups to include non-faithful actions on trees, linking them to automata and symbolic dynamics, and establishes conditions under which such groups are of finite type.

Contribution

It introduces a generalized framework for self-similar groups connected to tree automata and symbolic dynamics, and proves a new theorem extending Kitchens' work to these groups.

Findings

Various classes of self-similar groups do not coincide.

A sufficient condition for a sofic tree shift to be of finite type is provided.

Certain self-similar groups' closures are not Rabin-recognizable.

Abstract

We generalize the notion of self-similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but do not act faithfully on it. The elements of such a group correspond to labeled trees which may be recognized by a tree automaton (e.g. Rabin, B\"{u}chi, etc.), or considered as elements of a tree shift (e.g. of finite type, sofic) as in symbolic dynamics. We give examples to show that the various classes of self-similar groups defined in this way do not coincide. As the main result, extending the classical result of Kitchens on one-dimensional group shifts, we provide a sufficient condition for a self-similar group whose elements form a sofic tree shift to be a tree shift of finite type. As an application, we show that the closure of certain self-similar groups of tree automorphisms are not Rabin-recognizable. \end{abstract}