The virtual first Betti number of soluble groups

TL;DR

This paper proves that finitely presented groups with a specific nilpotent-by-abelian-by-finite structure have a uniform upper bound on the first Betti number across all their finite index subgroups.

Contribution

It establishes a new upper bound on the first Betti number for a class of finitely presented groups with a particular nilpotent-by-abelian-by-finite structure.

Findings

Bound on the first Betti number for finite index subgroups

Applicable to finitely presented nilpotent-by-abelian-by-finite groups

Advances understanding of group invariants in algebraic topology

Abstract

We show that if a group G is finitely presented and nilpotent-by-abelian-by-finite, then there is an upper bound on the first betti number of M as M runs through all subgroups of finite index in G.