The finite and solvable genus of finitely generated free and surface groups

Abstract

AbstractLet $${\mathcal {C}}$$ C be the pseudovariety $${\mathcal {F}}$$ F of all finite groups or the pseudovariety $${\mathcal {S}}$$ S of all finite solvable groups and let $$\Gamma $$ Γ be either a finitely generated free group or a surface group. The $${\mathcal {C}}$$ C -genus of $$\Gamma $$ Γ , denoted by $${\mathcal {G}}_{{\mathcal {C}}}(\Gamma )$$ G C ( Γ ) , consists of the isomorphism classes of finitely generated residually-$$\mathcal C$$ C groups G having the same quotients in $${\mathcal {C}}$$ C as $$\Gamma $$ Γ . We show that the groups from $${\mathcal {G}}_{{\mathcal {C}}}(\Gamma )$$ G C ( Γ ) are residually-p for all primes p. This answers a question of Gilbert Baumslag and shows that the groups in the genus are residually finite rationally solvable groups. This leads to a positive solution of particular case of a question of Alexander Grothendieck: if F is a free group, G is a finitely generated residually-$${\mathcal {C}}$$ C group and $$u:F\rightarrow G$$ u : F → G is a homomorphism such that the induced map of pro-$${\mathcal {C}}$$ C completions $$u_{\widehat{{\mathcal {C}}}} : F_{\widehat{{\mathcal {C}}}}\rightarrow G_{\widehat{{\mathcal {C}}}}$$ u C ^ : F C ^ → G C ^ is an isomorphism, then u is an isomorphism.