Two-generator hyperbolic orbifolds
by
Natalia Kopteva
Novosibirsk Technical University,Russia
Coauthors: Elena Klimenko (Novosibirsk Technical University,Russia)

The classification problem for hyperbolic 3-manifolds and orbifolds is one of the main problems in low-dimensional topology. This problem is equivalent to the classification problem for discrete subgroups of PSL(2, C). However, both problems are far from being completely solved.

It is interesting to consider two-generator groups among all subgroups of PSL(2, C), because, for example, two-generator groups uniformize the hyperbolic manifold and orbifold of minimal known volume.

In 1998 Gehring, Gilman and Martin [1] introduce the class of two-generator groups with real parameters. As parameters for \Gamma = <f, g> they take \beta = tr²f-4, \beta'=tr²g-4 and \gamma = tr^[f, g]-2. We denote GGM={\Gamma = <f, g>|f, g in PSL(2, C), \beta, \beta', \gamma in R}. To investigate the class GGM we have to consider all combinations of the following types of generators: elliptic, parabolic, hyperbolic and \pi-loxodromic. The most part of these cases was studied by E. Klimenko.

We consider the case when one of the generators is an elliptic element of even order and the other is a hyperbolic one, the axes of the generators intersect at an acute angle. The theorem which gives necessary and sufficient conditions for discreteness of such groups in terms of parameters has been proved.

The method of the proof is geometric. We construct a fundamental polyhedron for some group in which \Gamma has a finite index. Then we conclude whether or not \Gamma is discrete. And we obtain orbifolds by gluing equivalent points on the boundary of the fundamental polyhedron. So, we have new series of hyperbolic orbifolds. A list of all orbifolds that correspond to discrete groups in our case also will be presented.

References

[1] F. Gehring, J. Gilman, G. Martin, Discrete groups with real parameters, Preprint, 1998.

Date received: February 21, 2000