Liftable self-similar groups and scale groups

2023/12/09 by Rostislav Grigorchuk, Grigorchuk, Rostislav, Dmytro Savchuk +1
Computer Science · Mathematics · #20E08 #22D05 #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2312.05427

openalex publication_date 2023/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the (d+1)-regular tree \widetilde T_d+1, where d is the residual degree. Then we introduce the class of liftable self-similar groups acting on a d-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on \widetilde T_d+1 fixing one of the ends. The closures of these extensions in Aut(\widetilde T_d+1) are totally disconnected locally compact group that belong to the class of scale groups. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely or groups admitting finite L-presentations. In particular, we show that the finitely presented group constructed by the first author and the finitely presented HNN extension of the Basilica group embed into the group \mathcal D(\mathbb Q₂) of dilations of the field \mathbb Q₂ of 2-adic numbers. These actions, translated to \widetilde T₃, are 2-transitive on the punctured boundary of \widetilde T₃. Also we explore scale-invariant groups with the purpose of getting new examples of scale groups.

Related

• Scale Groups
• Extensions of automorphisms of self-similar groups
• Liftability and Contracting Property of Multi-EGS Groups
• Scale-multiplicative semigroups and geometry: automorphism groups of\n trees
• Intransitive Self-similar Groups