Agrarian and $\ell^2$-Betti numbers of locally indicable groups, with a twist

TL;DR

This paper establishes a relationship between twisted and usual $ ext{l}^2$-Betti numbers for locally indicable groups, provides formulas for $ ext{l}^2$-Betti numbers in fibrations, and introduces new inequalities and tools in agrarian invariants.

Contribution

It proves that twisted $ ext{l}^2$-Betti numbers equal the usual ones scaled by the representation dimension, answering a question of Lück, and develops the theory of generalized agrarian invariants.

Findings

Twisted $ ext{l}^2$-Betti numbers equal scaled usual $ ext{l}^2$-Betti numbers.

Formulas for $ ext{l}^2$-Betti numbers in fibrations with locally indicable base.

An inequality relating twisted Alexander and Thurston norms.

Abstract

We prove that twisted $\ell^2$-Betti numbers of locally indicable groups are equal to the usual $\ell^2$-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of L\"uck for this class of groups. It also leads to two formulae: given a fibration $E$ with base space $B$ having locally indicable fundamental group, and with a simply-connected fibre $F$, the first formula bounds $\ell^2$-Betti numbers $b_i^{(2)}(E)$ of $E$ in terms of $\ell^2$-Betti numbers of $B$ and usual Betti numbers of $F$; the second formula computes $b_i^{(2)}(E)$ exactly in terms of the same data, provided that $F$ is a high-dimensional sphere. We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and $3$-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.