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Surface groups within Baumslag doubles

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  19 January 2011

Benjamin Fine  and
Gerhard Rosenberger
Benjamin Fine
Affiliation:
Department of Mathematics, Fairfield University, Fairfield, CT 06430, USA ([email protected])
Gerhard Rosenberger
Affiliation:
Heinrich-Barth Strasse 1, 20146 Hamburg, Germany
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Abstract

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A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

Keywords

hyperbolic grouporientable surface groupquadratic word

MSC classification

Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 20F65: Geometric group theory 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E07: Subgroup theorems; subgroup growth
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 1 , February 2011 , pp. 91 - 97
Copyright
Copyright © Edinburgh Mathematical Society 2011

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