Virtual homology of limit groups and profinite rigidity of direct products

TL;DR

This paper investigates the virtual homology properties of residually free groups, develops techniques to recognize direct product decompositions, and proves profinite rigidity for certain classes of groups, advancing understanding of their algebraic and topological structures.

Contribution

It establishes the finiteness of the virtual second Betti number for specific residually free groups and proves profinite rigidity of direct products of free and surface groups.

Findings

Finite virtual second Betti number characterizes free, free abelian, and surface groups.

Profinite rigidity of direct products of free and surface groups among residually free groups.

Confirmation of Mel'nikov's surface group conjecture in the residually free setting.

Abstract

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, free abelian or the fundamental group of a closed surface. We also prove a similar statement in higher dimensions. We then develop techniques involving rank gradients of pro-$p$ groups, which allow us to recognise direct product decompositions. Combining these ideas, we show that direct products of free and surface groups are profinitely rigid among finitely presented, residually free groups, partially resolving a conjecture of Bridson's. Other corollaries that we obtain include a confirmation of Mel'nikov's surface group conjecture in the residually free case, and a description of closed aspherical manifolds of dimension at least $5$ with a residually free fundamental group.