Investigating self-similar groups using their finite $L$-presentation

TL;DR

This paper explores the use of finite L-presentations to analyze self-similar groups, enabling the application of algorithms for finitely presented groups to study their structure and properties.

Contribution

It provides an overview of algorithms for finitely L-presented groups and demonstrates their implementation for analyzing specific self-similar groups.

Findings

Implementation in computer algebra systems facilitates detailed structural analysis.

Applied to Fabrykowski-Gupta groups, revealing new insights.

Algorithms extend finitely presented group techniques to recursive presentations.

Abstract

Self-similar groups provide a rich source of groups with interesting properties; e.g., infinite torsion groups (Burnside groups) and groups with an intermediate word growth. Various self-similar groups can be described by a recursive (possibly infinite) presentation, a so-called finite $L$-presentation. Finite $L$-presentations allow numerous algorithms for finitely presented groups to be generalized to this special class of recursive presentations. We give an overview of the algorithms for finitely $L$-presented groups. As applications, we demonstrate how their implementation in a computer algebra system allows us to study explicit examples of self-similar groups including the Fabrykowski-Gupta groups. Our experiments yield detailed insight into the structure of these groups.