# Essential surfaces in graph pairs

## TL;DR

This paper proves the existence of surface subgroups in certain hyperbolic groups, reducing Gromov's question to rigid cases and showing that limit groups with the same profinite completion as free groups are actually free.

## Contribution

It establishes the presence of surface subgroups in fundamental groups of graphs of free groups with cyclic edge groups and in limit groups, advancing understanding of hyperbolic groups.

## Key findings

- Surface subgroups exist in fundamental groups of graphs of free groups with cyclic edge groups.
- Limit groups with the same profinite completion as free groups are free.
- Gromov's question reduces to the case of rigid groups under certain conditions.

## Abstract

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, Gromov's 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.