Abstract
For a handlebody of genus \(g\ge 6\) it is shown that every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians and, in consequence, it is geometric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brendle, T., Margalit, D.: Commensurations of the Johnson kernel. Geom. Topol. 8, 1361–1384 (2004)
Cho, S., Koda, Y., Seo, A.: Arc complexes, sphere complexes, and goeritz groups. Michigan Math. J. 65, 333–351 (2016)
Farb, B., Ivanov, N.: The Torelli geometry and its applications. Math. Res. Lett. 12, 293–301 (2005)
Farb, B., Margalit, D.: A Primer on Mapping Class Group. Princeton University Press, PrincetonPrinceton (2011)
Fomenko, A.T., Matveev, S.: Algorithmic and computer methods for three-manifolds. In: Mathematics and its Applications, 425. Kluwer Academic Publishers, Dordrecht, xii+334 pp (1997). ISBN: 0-7923-4770-6
Harvey, W.: Boundary structure of the modular group, Riemann surfaces and related topics. In: Proceedings of the 1978 Stony Brook Conference, State University New York, Stony Brook, N.Y., 1978, Ann. Math. Stud. 97, pp. 245–251. Princeton University. Press, Princeton (1981)
Hatcher, A.: On triangulations of surfaces. Topol. Appl. 40, 189–194 (1991)
Hensel, S.: A primer on handlebody groups. https://www.mathematik.uni-muenchen.de/~hensel/papers/hno4.pdf
Irmak, E., McCarthy, J.D.: Injective simplicial maps of the arc complex. Turk. J. Math. 34, 339–354 (2010)
Ivanov, N.V.: Automorphisms of complexes of curves and of Teichmuller spaces. Int. Math. Res. Not. 14, 651–666 (1977)
Johnson, D.: The structure of Torelli group II, a characterization of the group generated by twists on bounding curves. Topology 24, 113–126 (1985)
Korkmaz, M., Schleimer, S.: Automorphisms of the disk complex. arXiv:0910.2038v1
Korkmaz, M.: Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topol. Appl. 95(2), 85–111 (1999)
Masur, H., Minsky, Y.N.: Quasiconvexity in the curve complex. In: The Tradition of Ahlfors and Bers, III, pp. 309–320. Contemporary Mathematics, 355, American Mathematical Society., Providence (2004)
McCullough, D.: Virtually geometrically finite mapping class groups of \(3\)-manifolds. J. Differ. Geom. 33(1), 1–65 (1991)
Putman, A.: A note on the connectivity of certain complexes associated to surfaces. Enseign. Math. (2) 54(3–4), 287–301 (2008)
Wajnryb, B.: Mapping class group of a handlebody. Fund. Math. 158(3), 195–228 (1998)
Acknowledgements
The authors would like to thank the anonymous referee for very helpful comments and suggestions which resulted in significant improvements of this paper. One of them was the clarity of the conclusions of Lemmata 14 and 15 as well as his short and elegant proof of Lemma 14 which allowed us to omit a long and technical part of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Availability of data
All data supporting the conclusions of this article are included within the article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Charitos, C., Papadoperakis, I. & Tsapogas, G. On the complex of separating meridians in handlebodies. manuscripta math. 170, 581–606 (2023). https://doi.org/10.1007/s00229-022-01368-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-022-01368-0