Prosolvable rigidity of surface groups

Abstract

Surface groups are known to be the Poincar\'e Duality groups of dimension two since the work of Eckmann, Linnell and M\"uller. We prove a prosolvable analogue of this result that allows us to show that surface groups are profinitely (and prosolvably) rigid among finitely generated groups that satisfy $\mathrm{cd}(G)=2$ and $b_2^{(2)}(G)=0$. We explore two other consequences. On the one hand, we derive that if $u$ is a surface word of a finitely generated free group $F$ and $v\in F$ is measure equivalent to $u$ in all finite solvable quotients of $F$ then $u$ and $v$ belong to the same $\mathrm{Aut}(F)$-orbit. Finally, we get a partial result towards Mel'nikov's surface group conjecture. Let $F$ be a free group of rank $n\geq 3$ and let $w\in F$. Suppose that $G=F/\langle\!\langle w\rangle\!\rangle$ is a residually finite group all of whose finite-index subgroups are one-relator groups. Then $G$ is 2-free. Moreover, we show that if $H^2(G; \mathbb{Z})\neq 0$ then $G$ must be a surface group.