Resumen de Prosolvable rigidity of surface groups

Andrei Jaikin-Zapirain , Ismael Morales

• Surface groups are known to be the Poincaré duality groups of dimension two since the work of Eckmann, Linnell and Müller. 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 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 ∈ F is measure equivalent to u in all finite solvable quotients of F, then u and v belong to the same 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 3 and let w ∈ F. Suppose that G = F/w 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 H2(G;Z) = 0 then G must be a surface group.