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

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