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