Normal generation and $\ell^2$-betti numbers of groups

TL;DR

This paper investigates the relationship between the normal rank of groups and their first -Betti number, proposing a conjecture, proving it for certain groups, and constructing examples of simple groups with prescribed properties.

Contribution

It introduces a conjecture relating normal rank and -Betti number, proves it for limits of left-orderable amenable groups, and constructs new simple groups with specific -Betti numbers and ranks.

Findings

Conjecture that -Betti number does not exceed normal rank minus one for torsion-free groups.

Proof of the conjecture for limits of left-orderable amenable groups.

Construction of simple groups with prescribed -Betti numbers and ranks.

Abstract

The \emph{normal rank} of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first $\ell^2$-Betti number and conjecture that inequality $\beta_1^{(2)}(G)$ does not exceed normal rank minus 1 for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every $n\ge 2$ and every $\e>0$, we give an example of a simple group $Q$ (with torsion) such that $\beta_1^{(2)}(Q) \geq n-1-\epsilon$. These groups also provide examples of simple groups of rank exactly $n$ for every $n\ge 2$; existence of such examples for $n> 3$ was unknown until now.