Essential surfaces in graph pairs

Abstract

A well-known question of Gromov asks whether every one-ended hyperbolic group Γ \Gamma has a surface subgroup. We give a positive answer when Γ \Gamma is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov’s question is reduced (modulo a technical assumption on 2-torsion) to the case when Γ \Gamma is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.