Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T21:15:10.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphisms of Fuchsian Groups and their Lifts to Free Groups

Published online by Cambridge University Press:  20 November 2018

Martin Lustig ,
Yoav Moriah  and
Gerhard Rosenberger
Martin Lustig
Affiliation:
Ruhr-Universität Bochum, Bochum, West Germany
Yoav Moriah
Affiliation:
Technion, Haifa, Israel
Gerhard Rosenberger
Affiliation:
Universität Dortmund, Dortmund, West Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper has been motivated by earlier work of the first two authors (see [3] ), where distinct Nielsen classes of generating systems for a Fuchsian group have been established and, in the case of odd and pairwise relative prime exponents π(i),classified. As a consequence they could distinguish nonisotopic Heegaard decompositions of Seifert fibred 3-manifolds. In proving that these decompositions are actually non-homeomorphic (see [3], Section 2), they investigated the question whether the different Nielsen classes of generating systems for G remain distinct, if one passes over to the weaker notion of “Nielsen equivalence up to automorphisms” (see [12], p. 3.5, 4.11 a-c): this means that the automorphisms of G are added to the Nielsen equivalence relations on the generators.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 1 , 01 February 1989 , pp. 123 - 131
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Kalia, R.N. and Rosenberger, G., Über Untergruppen ebener diskontinuierlicher Gruppen, Contemporary Mathematics 33 (1984), 308–327.Google Scholar
2. Lustig, M., Nielsen equivalence and simple-homotopy type, preprint.Google Scholar
3. Lustig, M. and Moriah, Y., Nielsen equivalence in Fuchsian groups and Seifert fibre spaces, preprint.Google Scholar
4. Lyndon, R.C. and Schupp, P.E., Combinatorial group theory, Modern Surveys in Math. 89 (Springer-Verlag, 1977).Google Scholar
5. Peczynski, N., Über Erzeugendensysteme von Fuchsschen Gruppen, Ph.D. Thesis, Bochum (1975).Google Scholar
6. Peczynski, N., Rosenberger, G. and Zieschang, H., Über Erzeugende ebener diskontinuierlicher Gruppen, Inventiones Math. 29 (1975), 161–180.Google Scholar
7. Rosenberger, G., Zum Rang- und Isomorphieproblem von freien Produkten mit Amalgam, Habilitationsschrift, Hamburg (1974).Google Scholar
8. Rosenberger, G., Automorphismen ebener diskontinuierlicher Gruppen, in Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference, Annals of Math. Studies 97, (Princeton Univ. Press, 980).Google Scholar
9. Rosenberger, G., All generating pairs of all two-generator Fuchsian groups, Arch. Math. 46 (1986), 198–204.Google Scholar
10. Rosenberger, G., Minimal generating systems for plane discontinuous groups and an equation in free groups, preprint.Google Scholar
11. Zieschang, H., Uber Automorphismen ebener diskontinuierlicher Gruppen, Math. Annalen 766(1966), 148–167.Google Scholar
12. Zieschang, H., Über die Nielsensche Kurzungsmethode in freien Gruppen mit Amalgam, Inventiones Math. 10 (1970), 4–37.Google Scholar