Virtual First Betti Number of GGS Groups

TL;DR

This paper establishes a criterion for groups to have zero virtual first Betti number and provides numerous examples of specific groups that are torsion-free, finitely generated, residually finite, but not virtually diffuse, addressing an open question.

Contribution

It introduces a new criterion for vanishing virtual first Betti number and constructs examples of groups with particular properties that answer an existing open question.

Findings

Groups with vanishing virtual first Betti number identified

Constructed examples of torsion-free, finitely generated, residually finite groups not virtually diffuse

Answered an open question by Kionke and Raimbault

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.