Geometriae Dedicata

, Volume 110, Issue 1, pp 159–190 | Cite as

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

Article

Abstract

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ‘hyperelliptic’ three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.

Keywords

Fuchsian groups Kleinian groups Schottky groups Riemann surfaces hyper elliptic surfaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlfors, L. 1979Complex Analysis3McGraw-HillNew YorkGoogle Scholar
  2. 2.
    Bers, L. 1976Nielsen extensions of Riemann surfacesAnn. Acad. Sci Fenn22934Google Scholar
  3. 3.
    Fenchel, W. 1989Elementary Geometry in Hyperbolic Space Stud. Math. 11De GruyterBerlinGoogle Scholar
  4. 4.
    Gilman, J. 1988Inequalities and discretenessCanad. J. Math.40115130Google Scholar
  5. 5.
    Gilman, J. 1995Two generator discrete subgroups of PSL(2, \(\cal{R}\))Mem. Amer. Math. Soc.117561Google Scholar
  6. 6.
    Gilman, J. and Keen, L.: Word Sequences and Intersection Numbers. Complex Manifolds and Hyperbolic Geometry (Guanajuato 2001), Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 231–249.Google Scholar
  7. 7.
    Gilman, J., Maskit, B. 1991An algorithm for two-generator discrete groupsMich. Math. J.381332Google Scholar
  8. 8.
    Keen, L.: Canonical polygons for finitely generated fuchsian groups, Acta Math. 115 (1966).Google Scholar
  9. 9.
    Keen, L. 1966Intrinsic moduli on Riemann surfacesAnn. Math.84404420Google Scholar
  10. 10.
    Kapovich, M.: Hyperbolic Manifolds and Discrete Groups, Birkhäuser Basel, 2001.Google Scholar
  11. 11.
    Knapp, A. W. 1968Doubly generated Fuchsian groupsMich. Math. J.15289304Google Scholar
  12. 12.
    Hurwitz, A. 1893Algebraische Gebilde mit eindeutigen Tranformationen in sicMath. Ann.41409442Google Scholar
  13. 13.
    Maskit, B. 1988Kleinian GroupsSpringer-VerlagNew YorkGoogle Scholar
  14. 14.
    Matelski, P. 1982The classification of discrete two-generator subgroups of PSL(2, \(\cal{R}\))Israel J. Math.42309317Google Scholar
  15. 15.
    Purzitsky, N. and Rosenberger, G.: All two-generator fuchsian groups, Math. Z128 (1972), 245–251. (Correction: Math. Z.132(1973) 261–262.)Google Scholar
  16. 16.
    Purzitsky, N. 1976All two-generator Fuchsian groupsMath. Z.1478792Google Scholar
  17. 17.
    Rosenberger, G. 1986All generating pairs of all two-generator fuchsian groupsArch. Mat.46198204Google Scholar
  18. 18.
    Springer, G. 1957Introduction to Riemann SurfacesAddison-WesleyReading, MAGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics Rutgers UniversityNewarkU.S.A.
  2. 2.Mathematics DepartmentCUNY Lehman College and Graduate CenterBronxU.S.A.

Personalised recommendations