Essential surfaces in graph pairs

Wilton, Henry

doi:10.1090/jams/901

Essential surfaces in graph pairs
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by Henry Wilton

J. Amer. Math. Soc. 31 (2018), 893-919

DOI: https://doi.org/10.1090/jams/901

Published electronically: June 18, 2018

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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.

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