Abstract
For aficionados of fundamental polyhedra in the study of the kleinian groups, it is helpful to be able to choose in each particular case a polyhedron with the simplest possible local structure about its edges and vertices. For example, in the study of small deformations as in [4], a fundamental polyhedron for one group is compared to those of nearby groups; if the one polyhedron is as simple as possible, the nearby ones will tend to be as well. It is the purpose of the present note to find such polyhedra. Indeed, we will show that the “generic” fundamental polyhedra for a given group are as simple as the algebraic/geometric structure of the group allows. When the group has no elliptic transformations, the features of the generic polyhedra will be precisely identified. In groups with torsion, on the other hand, certain configurations of elliptic transformations, for example three elliptics whose axes are pairwise coplanar, involve additional difficulties and we have decided to leave these cases aside.
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© 1988 Springer-Verlag New York Inc.
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Jorgensen, T., Marden, A. (1988). Generic fundamental polyhedra for kleinian groups. In: Drasin, D., Earle, C.J., Gehring, F.W., Kra, I., Marden, A. (eds) Holomorphic Functions and Moduli II. Mathematical Sciences Research Institute Publications, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9611-6_7
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DOI: https://doi.org/10.1007/978-1-4613-9611-6_7
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