On the top-dimensional $\ell^2$-Betti numbers

TL;DR

This paper introduces a technique linking the non-vanishing of top-dimensional $ ell^2$-Betti numbers of actions to those of sub-actions, with applications to automorphism groups, 3-manifold groups, and product groups.

Contribution

It presents a new trick to relate top-dimensional $ ell^2$-Betti numbers of actions to sub-actions, leading to several new vanishing and non-vanishing results.

Findings

Non-vanishing of $ ell^2$-Betti numbers for Aut($F_n$) and Out($F_n$) in their virtual cohomological dimension.

Vanishing of $ ell^2$-Betti numbers in degree 3 and 2 for subgroups of 3-manifold groups.

Determination of ergodic dimension for groups like $F_2^d imes Z$ as $d + 1$.

Abstract

The purpose of this note is to introduce a trick which relates the (non)-vanishing of the top-dimensional $\ell^2$-Betti numbers of actions with that of sub-actions. We provide three different types of applications: we prove that the $\ell^2$-Betti numbers of Aut($F_n$) and Out($F_n$) (and of their Torelli subgroups) do not vanish in degree equal to their virtual cohomological dimension, we prove that the subgroups of the 3-manifold groups have vanishing $\ell^2$-Betti numbers in degree 3 and 2 and we prove for instance that $F_2^d \times Z$ has ergodic dimension $d + 1$.