On the simplicity of Nekrashevych algebras of contracting self-similar groups

TL;DR

This paper establishes necessary and sufficient conditions for the simplicity of Nekrashevych algebras of contracting self-similar groups, providing an algorithm to determine simplicity over various fields and analyzing several well-known groups.

Contribution

It introduces a complete characterization of simplicity for Nekrashevych algebras of contracting groups and provides an explicit algorithm for field characteristic determination.

Findings

Determined simplicity conditions for Nekrashevych algebras of various groups.

Developed an algorithm to identify fields where the algebra is simple.

Applied methods to groups like the Basilica, Grigorchuk, and Gupta-Sidki groups.

Abstract

Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych $C^\ast$-algebras. In particular, Nekrashevych, Clark, Exel, Pardo, Sims and Starling have studied the question of simplicity of Nekrashevych algebras, in part, because non-simplicity of the complex algebra implies non-simplicity of the $C^\ast$-algebra. In this paper we give necessary and sufficient conditions for the Nekrashevych algebra of a contracting group to be simple. Nekrashevych algebras of contracting groups are finitely presented. We give an algorithm which on input the nucleus of the contracting group, outputs all characteristics of fields over which the corresponding Nekrashevych algebra is simple. Using our methods, we determine the fields over which the Nekrashevych algebras of a number of well-known contracting groups are simple including the Basilica group, Gupta-Sidki groups, GGS-groups, multi-edge spinal groups, \v{S}uni\'{c} groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.