Algorithms for finitely L-presented groups and their applications to some self-similar groups

作者：Hartung Rene^*

来源：Expositiones Mathematicae, 2013, 31(4): 368-384.

DOI：10.1016/j.exmath.2013.07.001

• Recursive presentations
• Self-similar groups
• Grigorchuk group
• Fabrykowski-Gupta groups
• Coset enumeration
• Finite index subgroups
• Reidemeister-Schreier theorem
• Nilpotent quotients
• Solvable quotients

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 and the Grigorchuk group. Our experiments yield detailed insight into the structure of these groups.

• 出版日期2013