On finite generation of self-similar groups of finite type

TL;DR

This paper establishes criteria to analyze self-similar groups of finite type, focusing on their finiteness, transitivity, and finite generation, supported by computational results for binary trees.

Contribution

It introduces new criteria for classifying self-similar groups of finite type and provides computational evidence for their properties at specific depths.

Findings

No infinite topologically finitely generated groups at depth 3 for binary trees.

Existence of 32 such groups at depth 4.

Criteria help determine finiteness and generation properties.

Abstract

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth $3$, and there are $32$ such groups for depth $4$.