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We show that the virtual second Betti number of a finitely generated,residually free group $G$ is finite if and only if $G$ is either free, freeabelian or the fundamental group of a closed surface. We continue and calculatethe virtual Betti numbers of limit groups in all dimensions. We also developtechniques involving rank gradients of pro-$p$ groups, which allow us torecognize direct product decompositions. Combining the above ideas, we showthat direct products of free and surface groups are profinitely rigid amongfinitely presented, residually free groups, partially resolving a conjecture ofBridson's. Other corollaries that we obtain include a confirmation ofMel'nikov's surface group conjecture in the residually free case, and adescription of closed aspherical manifolds of dimension at least $5$ with aresidually free fundamental group.

Virtual homology of residually free groups and profinite rigidity of direct products