Virtual First Betti Number of GGS Groups

TL;DR

The paper addresses the question of whether torsion-free, finitely generated, residually finite groups that are not virtually diffuse exist, and provides a criterion for vanishing virtual first Betti number in groups with all finite quotients of order prime to a fixed prime $q$. It then instantiates this framework in Grigorchuk-Gupta-Sidki (GGS) groups acting on $p$-ary rooted trees, producing infinite families of groups whose finite quotients are all $p$-groups and are not virtually diffuse. Two independent proofs are given for non-virtual-diffuseness: a branching-based argument leveraging the infinite direct sum structure inside, and a congruence subgroup property (CSP) based approach showing the derived subgroup is torsion-free and not virtually locally indicable. These results answer a question of Kionke and Raimbault and illuminate connections between amenability, CSP, virtual diffuseness, and group-ring properties in this class of groups.

Abstract

We observe a criterion for groups to have vanishing virtual first Betti number and use it to give infinitely many examples of torsion-free, finitely generated, residually finite groups which aren't virtually diffuse. This answers a question raised by Kionke and Raimbault.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Theorem 3
• Lemma 4
• proof
• Definition 5
• Definition 6
• Theorem 7
• Definition 8
• Definition 9