On groups which are not finitely defined in every quasivariety of groups

We continue to study quasivarieties of groups closed under direct Z-wreath products. We show that such quasivarieties contain finitely generated groups which are not finitely defined in every quasivariety of groups. We establish the existence of continuum many finitely generated groups every of which is not finitely defined in each quasivariety of groups. We construct the group which is finitely defined in the class of all torsion-free groups and is not finitely defined in the class of all groups.