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Journal of the American Mathematical Society

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Essential surfaces in graph pairs


Author: Henry Wilton
Journal: J. Amer. Math. Soc. 31 (2018), 893-919
MSC (2010): Primary 20F65, 20F67, 57M07
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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Additional Information

Henry Wilton
Affiliation: DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: h.wilton@maths.cam.ac.uk

DOI: https://doi.org/10.1090/jams/901
Received by editor(s): January 15, 2017
Received by editor(s) in revised form: March 14, 2018
Published electronically: June 18, 2018
Additional Notes: The author was supported by EPSRC Standard Grant EP/L026481/1.
Article copyright: © Copyright 2018 American Mathematical Society

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