STRUCTURES OF GEOMETRIC QUOTIENT ORBIFOLDS OF THREE-DIMENSIONAL G-MANIFOLDS OF GENUS TWO

- Journal title : Journal of the Korean Mathematical Society
- Volume 46, Issue 4, 2009, pp.859-893
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2009.46.4.859

Title & Authors

STRUCTURES OF GEOMETRIC QUOTIENT ORBIFOLDS OF THREE-DIMENSIONAL G-MANIFOLDS OF GENUS TWO

Kim, Jung-Soo;

Kim, Jung-Soo;

Abstract

In this article, we will characterize structures of geometric quotient orbifolds of G-manifold of genus two where G is a finite group of orientation preserving diffeomorphisms using the idea of handlebody orbifolds. By using the characterization, we will deduce the candidates of possible non-hyperbolic geometric quotient orbifolds case by case using W. Dunbar`s work. In addition, if the G-manifold is compact, closed and the quotient orbifold`s geometry is hyperbolic then we can show that the fundamental group of the quotient orbifold cannot be in the class D.

Keywords

orbifold;finite group action;handlebody orbifold;Heegaard splitting;

Language

English

Cited by

References

1.

M. Boileau and H. Zieschang, Nombre de ponts et generateurs meridiens des entrelacs de Montesinos, Comment. Math. Helv. 60 (1985) 270–279

2.

W. D. Dunbar, Geometric orbifolds, Rev. Mat. Univ. Complut. Madrid 1 (1988), no. 1-3, 67–99

3.

E. Klimenko and N. Kopteva, Two-generator Kleinian orbifolds, arXiv:math/0606066

4.

T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984), no. 2, 437–455

5.

D. McCullough, A. Miller, and B. Zimmermann, Group actions on handlebodies, Proc. London Math. Soc. (3) 59 (1989), no. 2, 373–416

6.

W. H. Meeks and S.-T. Yau, The equivariant loop theorem for three-dimensional manifolds and a review of the existence theorems for minimal surfaces, The Smith conjecture (New York, 1979), 153–163, Pure Appl. Math., 112, Academic Press, Orlando, FL, 1984

7.

A. Miller and B. Zimmermann, Large groups of symmetries of handlebodies, Proc. Amer. Math. Soc. 106 (1989), no. 3, 829–838

8.

J. P. Serre, Trees, Translated from the French original by John Stillwell. Corrected 2nd printing of the 1980 English translation., Springer-Verlag, Berlin, 2003