$L^2$-Betti numbers and non-unitarizable groups without free subgroups

TL;DR

This paper constructs examples of non-unitarizable groups lacking free subgroups and explores their properties, including torsion groups with non-zero $L^2$-Betti numbers, linking group theory and $L^2$-invariants.

Contribution

It provides the first known examples of such non-unitarizable groups without free subgroups and connects $L^2$-Betti numbers to hyperbolic group residual finiteness.

Findings

Existence of non-unitarizable groups without free subgroups

Construction of torsion groups with non-zero $L^2$-Betti numbers

Relation between hyperbolic groups and $L^2$-Betti number approximation

Abstract

We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with non-vanishing first $L^2$-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of $L^2$-Betti numbers.