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Surface groups within Baumslag doubles

Part of: Structure and classification of infinite or finite groups Special aspects 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 (fine@fairfield.edu)
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

References

1.Ackermann, P., Fine, B. and Rosenberger, G., Surface groups: motivating examples in combinatorial group theory, in Groups St Andrews 2005, London Mathematical Society Lecture Notes Series, Volume 339, pp. 96–130 (Cambridge University Press, 2007).Google Scholar
2.Baumslag, G., On generalised free products, Math. Z. 78 (1962), 423–438.CrossRefGoogle Scholar
3.Bestvina, M. and Feighn, M., A combination theorem for negatively curved groups, J. Diff. Geom. 35 (1992), 85–101.Google Scholar
4.Comerford, L., Edmonds, C. and Rosenberger, G., Commutators as powers in free products, Proc. Am. Math. Soc. 122 (1994), 47–52.Google Scholar
5.Fine, B., Rosenberger, G. and Stille, M., Nielsen transformations and applications: a survey, in Groups Korea 94 (De Gruyter, Berlin, 1995).Google Scholar
6.Gordon, C. and Wilton, H., Surface subgroups of doubles of free groups, in preparation.Google Scholar
7.Juhasz, A. and Rosenberger, G., On the combinatorial curvature of groups of F-type and other one-relator products of cyclics, Contemp. Math. 169 (1994), 373–384.CrossRefGoogle Scholar
8.Kharlampovich, O. and Myasnikov, A., Hyperbolic groups and free constructions, Trans. Am. Math. Soc. 350(2) (1998), 571–613.Google Scholar
9.Kim, S. and Wilton, H., Surface subgroups of doubles of free groups, in preparation.Google Scholar
10.Lyndon, R. and Schupp, P. E., Combinatorial group theory (Springer, 1977).Google Scholar
11.Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (Wiley Interscience, 1966).Google Scholar
12.Rosenberger, G., On one-relaor groups that are free products of two free groups with cyclic amalgamation, in Groups St Andrews 1981, pp. 328–344 (Cambridge University Press, 1981).Google Scholar
13.Zieschang, H., Über die Nielsensche Küerzungmethode in freien Produkten mit Amalgam, Invent. Math. 10 (1970), 4–37.CrossRefGoogle Scholar