2020/08/10 by Benjamin Steinberg, Steinberg, Benjamin, Nóra Szakács +1 · 6 citations
Mathematics · #16S88 #20F65 #20M18 #20M25 #47L40 #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #advanced mathematical theories #math.GR #math.OA #math.RA #msc:16S88 #msc:20F65 #msc:20M18 #msc:20M25 #msc:47L40
paper · pdf · doi:10.48550/arxiv.2008.04220
We improved exposition, added a discussion of the Hausdorff condition for groupoids of contracting groups and removed some examples
openalex publication_date 2020/08/10 · arxiv created 2021/07/06 · arxiv updated 2021/07/07 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28
Nekrashevych algebras of self-similar group actions are natural generalizations of the classical Leavitt algebras. They are discrete analogues of the corresponding Nekrashevych C^∗-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^∗-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, Šunić groups associated to polynomials (this latter family includes the Grigorchuk group, Grigorchuk-Erschler group and Fabrykowski-Gupta group) and self-replicating spinal automaton groups.