Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T01:05:01.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometrically finite kleinian groups and parabolic elements

Published online by Cambridge University Press:  20 January 2009

Ken'ichi Ohshika
Ken'ichi Ohshika
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 1 , February 1998 , pp. 141 - 159
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Abikoff, W., On boundaries of Teichmüller spaces and on Kleinian groups III, Acta Math. 134 (1975), 211–237.Google Scholar
2.Ahlfors, L., Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–423.CrossRefGoogle Scholar
3.Bers, L., Spaces of Kleinian groups (Lecture Notes on Mathematics 155, 1970), 9–34.Google Scholar
4.Bonahon, F., Bouts de variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71–158.Google Scholar
5.Bonahon, F., Cobordism of automorphisms of surfaces, Ann. Sci. Ecole Norm. Sup. (4) 16 (1983), 237–270.Google Scholar
6.Canary, R. D., Epstein, D. B. A. and Green, P., Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic spaces (London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, 1987), 3–92.Google Scholar
7.Culler, M. and Shalen, P., Variety of group representations and splittings of 3-manifolds, Ann. of Math. 117 (1983), 109–146.Google Scholar
8.Epstein, D. B. A. and Marden, A., Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, in Analytical and geometric aspects of hyperbolic spaces (London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, 1987), 113–253.Google Scholar
9.Hempel, J., 3-Manifolds (Ann. of Math. Studies, No. 86. Princeton Univ. Press).Google Scholar
10.Jaco, W., Lectures on three-manifold topology (CBMS regional conf. ser. in math. 43, 1980).CrossRefGoogle Scholar
11.Jaco, W. and Shalen, P., Seifert fibered spaces in 3-manifolds (Mem. Amer. Math. Soc. 21, 1979).Google Scholar
12.Jørgensen, T., On discrete groups of Möbius transformations, Amer. J. Math. 98 (1976), 739–749.Google Scholar
13.Jørgensen, T. and Marden, A., Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47–72.CrossRefGoogle Scholar
14.Marden, A., The geometry of finitely generated Kleinian groups, Ann. of Math. 99 (1974), 465–496.Google Scholar
15.Maskit, B., Parabolic elements on Kleinian groups, Ann. of Math. 117 (1983), 659–668.Google Scholar
16.Maskit, B., Comparison of hyperbolic length and extremal length, Ann. Acad. Sci. Fern. Ser. A I Math. 10 (1985), 381–386.CrossRefGoogle Scholar
17.Maskit, B., Kleinian groups (Grundlehren der mathematischen Wissenschaften 287, Springer-Verlag, Berlin-Heidleberg, 1988).Google Scholar
18.Mccullough, D., Compact submanifolds with boundary, Quart. J. Math. Oxford Ser. (2) 37 (1986), 299–307.CrossRefGoogle Scholar
19.Meeks, W. III and Yau, S. T., The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn's lemma, Topology 21 (1982), 409–442.Google Scholar
20.Mccullough, D., Miller, A. and Swarup, G. A., Uniqueness of cores of non-compact 3-manifolds, J. London Math. Soc. 32 (1985), 548–556.CrossRefGoogle Scholar
21.Morgan, J., On Thurston's uniformization theorem for three-dimensional manifolds, in The Smith Conjecture (Academic Press, New York, London, 1984), 37–125.Google Scholar
22.Morgan, J. and Shalen, P., Degeneration of hyperbolic structures III, action of 3-manifold groups on trees and Thurston's compactness theorem, Ann. Math. 120 (1988), 457–519.Google Scholar
23.Ohshika, K., Ending laminations and boundaries for deformation spaces of Kleinian groups, J. London Math. Soc. (2) 42 (1990), 111–121.CrossRefGoogle Scholar
24.Ohshika, K., Geometric behaviour of Kleinian groups on boundaries for deformation spaces, Quart. J. Math. Oxford (2) 43 (1992), 97–111.CrossRefGoogle Scholar
25.Ohshika, K., Parabolization of elements of Kleinian groups, in Topology and Teichmüller space (World Scientific, 1996), 221–236.CrossRefGoogle Scholar
26.Scott, G. P., Compact submanifolds of 3-manifolds, J. London Math. Soc. (2) 7 (1973), 246–250.Google Scholar
27.Shapiro, A. and Whitehead, J. H. C., A proof and extension of Dehn's lemma, Bull. Amer. Math. Soc. 64 (1958), 174–178.Google Scholar
28.Stallings, J., On the loop theorem, Ann. of Math. 72 (1960), 12–19.CrossRefGoogle Scholar
29.Sullivan, D., On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann Surfaces and Related Topics (Ann. Math. Studies 97, Princeton Univ. Press, 1980), 465–498.Google Scholar
30.Sullivan, D., A finiteness theorem for cusps, Acta Math. 147 (1980), 289–299.CrossRefGoogle Scholar
31.Thurston, W., The geometry and topology of 3-manifolds (Lecture notes, Princeton Univ.).Google Scholar
32.Thurston, W., Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds, Ann. of Math. 124 (1986), 203–246.Google Scholar
33.Thurston, W., Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle, preprint, Princeton Univ.Google Scholar
34.Thurston, W., Hyperbolic structures on 3-manifolds III: Deformations of 3-manifolds with incompressible boundary, preprint, Princeton Univ.Google Scholar
35.Waldhausen, F., On irreducible 3-manifolds which are sufficiently large, Ann. of Math. 87 (1968), 56–88.Google Scholar