# Infinite finitely generated non-amenable groups

I am interested if there is an example of an infinite finitely generated non-amenable group that is residually finite but does not contain non-abelian free subgroups.

What examples of infinite finitely gnerated perfect (non-simple) gropus are non-amenable but do not contain free subgroups?

Many thanks, Elisabeth

• Good question! The known examples of f.g. non-amenable group without free subgroups are Olshanskii monsters, Burnside groups and variants, as well as the more recent Monod examples. As far as I know, these groups are not, or are not expected, to be residually finite. So I guess the question is open. – YCor Sep 6 '13 at 10:20
• @Yvex, finitely generated infinite groups of bounded exponent are never residually finite by Zelmanov's solution of the restricted Burnside problem. – Benjamin Steinberg Sep 6 '13 at 12:52
• I now remember that Question 14 of my paper with Mann (2006, normalesup.org/~cornulier/law_rf4.pdf) includes the following closely related open question: Does there exist a finitely generated, residually finite group, that is not amenable and satisfies a nontrivial group law? (Of course it's even more difficult if we expect a positive answer.) – YCor Sep 6 '13 at 12:53
• @Benjamin: yes I know: I said "are not, or are not expected, to be RF", to mean that some of these groups are known not to be RF and some are expected not to be RF (e.g. Monod's groups, although it's just a rough intuition). – YCor Sep 6 '13 at 12:57

Osin and Luck give examples of infinite finitely generated residually finite torsion groups with positive first $\ell^2$-Betti number (and hence non-amenable) in the paper "Approximating the first L2-Betti number of residually finite groups", J. Topol. Anal. 3 (2011), no. 2, 153–160.