Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T04:19:35.686Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed points of imaginary reflections on hyperbolic handlebodies

Published online by Cambridge University Press:  28 September 2009

RUBEN A. HIDALGO  and
BERNARD MASKIT
RUBEN A. HIDALGO
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile. e-mail: [email protected]
BERNARD MASKIT
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook NY 11794-3651, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A Klein–Schottky group is an extended Kleinian group, containing no reflections and whose orientation-preserving half is a Schottky group. A dihedral-Klein–Schottky group is an extended Kleinian group generated by two different Klein–Schottky groups, both with the same orientation-preserving half. We provide a structural description of the dihedral-Klein–Schottky groups.

Let M be a handlebody of genus g, with a Schottky structure. An imaginary reflection τ of M is an orientation-reversing homeomorphism of M, of order two, whose restriction to its interior is an hyperbolic isometry having at most isolated fixed points. It is known that the number of fixed points of τ is at most g + 1; τ is called a maximal imaginary reflection if it has g + 1 fixed points. As a consequence of the structural description of the dihedral-Klein–Schottky groups, we are able to provide upper bounds for the cardinality of the set of fixed points of two or three different imaginary reflections acting on a handlebody with a Schottky structure. In particular, we show that maximal imaginary reflections are unique.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 135 - 158
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Agol, I. Tameness of hyperbolic 3-manifolds. arXiv:math.GT/0405568 (2004).Google Scholar
[2]Bujalance, E. and Singerman, D.The symmetry type of a Riemann surface. Proc. London Math. Soc. (3) 51 (1985), 501–519.CrossRefGoogle Scholar
[3]Bujalance, E., Costa, A. F. and Singerman, D.Application of Hoare's theorem to symmetries of Riemann surfaces. Ann. Acad. Scie. Fenn. Ser. A. I. Mathematica 18 (1993), 307–322.Google Scholar
[4]Calegari, D. and Gabai, D.Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Amer. Math. Soc. 19 No. 2 (2006), 385–446.CrossRefGoogle Scholar
[5]Gromadzki, G., Hidalgo, R. A. and Maskit, B. Symmetries of handlebodies with Schottky structure. Preprint.Google Scholar
[6]Harnack, A.Über die Vielteiligkeit der ebeunen algebraischen Kurven. Math. Ann. 10 (1876), 189–199.CrossRefGoogle Scholar
[7]Hidalgo, R. A. and Maskit, B.On Klein–Schottky groups. Pacific J. Math. 220 (2005), 313–328.CrossRefGoogle Scholar
[8]Izquierdo, M. and Singerman, D.Pairs of symmetries of Riemann surfaces. Ann. Acad. Sci. Fenn. Mathematica 23 (1998), 3–24.Google Scholar
[9]Maskit, B.Kleinian Groups. (GMW, Springer-Verlag, 1987).CrossRefGoogle Scholar
[10]Maskit, B.On Klein's combination theorem IV. Trans. Amer. Math. Soc. 336 (1993), 265–294.Google Scholar
[11]Maskit, B.Remarks on m-symmetric Riemann surfaces. Lipa's legacy (New York, 1995), Contemp. Math. 211 (1997), 443–445.Google Scholar
[12]Natanzon, S. M.Automorphisms of the Riemann surface of an M-curve. (Russian) Funktsional. Anal. i Prilozhen. 12, No. 3 (1978), 82–83.Google Scholar
[13]Natanzon, S. M.Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves. Trans. Moscow Math. Soc. 51 (1989), 1–51.Google Scholar