A well known question of Gromov asks whether every one-ended hyperbolic group $Gamma$ has a surface subgroup. We give a positive answer when $Gamma$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromovs question is reduced (modulo a technical assumption on 2-torsion) to the case when $Gamma$ 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.
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