Conformal Geometry and Dynamics

Author(s): Jouni Parkkonen
Journal: Conform. Geom. Dyn. 8 (2004), 143-157.
MSC (2000): Primary 30F40, 30F60
Posted: October 14, 2004
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Abstract: We study the geometric limits of sequences of loxodromic cyclic groups which arise from conformal Dehn surgery. The results are applied to obtain an asymptotic description of the shape of the main cusp of the Maskit embedding of the Teichmüller space of once-punctured tori.

1.: L. Bers, Spaces of Kleinian groups in Several Complex Variables I, Maryland 1970 Springer-Verlag, New York, Heidelberg, Berlin, 1970, 9-34. MR 0271333 (42:6216)
2.: L. Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 131-172. MR 0621883 (82k:32050)
3.: T. Drumm and J. Poritz, Ford and Dirichlet domains for cyclic subgroups of $\operatorname{PSL}_2\mathbb C$ acting on $\mathbb H^3_{\mathbb R}$ and $\partial\mathbb H^3_{\mathbb R}$ , Conformal Geometry and Dynamics 3 (1999), 116-150. MR 1716572 (2001c:30041)
4.: N. Gusevski ${\u{\i}}\kern.15em$ , On double generated Kleinian groups (Russian), Sibirsk. Mat. Zh. 23 (1982), 183-186. MR 0651891 (83f:30040)
5.: T. Jørgensen, On cyclic groups of Möbius transformations, Math. Scand. 33 (1973) 250-260. MR 0348103 (50:601)
6.: T. Jørgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), 47-72. MR 1060898 (91f:30068)
7.: L. Keen and C. Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), 719-749. MR 1241870 (95g:32030)
8.: S. Kerckhoff and W. Thurston, Non-continuity of the action of the modular group at Bers' boundary of Teichmüller space, Invent. Math. 100 (1990), 25-47. MR 1037141 (91a:57038)
9.: I. Kra, Canonical mappings between Teichmüller spaces, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 143-179. MR 0598682 (82b:32036)
10.: I. Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I: Teichmüller and Riemann spaces of Kleinian groups, Jour. Amer Math. Soc. 3(3) (1990), 500-578. MR 1049503 (91c:32014)
11.: B. Maskit, Moduli of marked Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 773-777. MR 0346149 (49:10875)
12.: B. Maskit, Kleinian Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1987. MR 0959135 (90a:30132)
13.: K. Matsuzaki and M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford University Press Oxford, 1998. MR 1638795 (99g:30055)
14.: C. T. McMullen, Cusps are dense, Ann. of Math. 133 (1991), 217-247. MR 1087348 (91m:30058)
15.: C. T. McMullen, Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups, J. Differential Geometry 51 (1999), 471-515. MR 1726737 (2001c:37045)
16.: Y. Minsky, The classification of punctured torus groups, Ann. of Math. 149 (1999), 559-626. MR 1689341 (2000f:30028)
17.: H. Miyachi, On the horocyclic coordinate for the Teichmüller space of once-punctured tori, Proc. Amer. Math. Soc. 130 (2002), 1019-1029. MR 1873775 (2002j:32010)
18.: H. Miyachi, Cusps in complex boundaries of one-dimensional Teichmüller spaces, Conform. Geom. Dyn. 7 (2003), 103-151 (electronic). MR 2023050 (2004j:30091)
19.: J. Parkkonen, Geometric limits of cyclic groups and the shape of Schottky space, Math. Proc. Cambridge Philos. Soc. 137 (2004), 55-68. MR 2075042
20.: W. P. Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University, 1979.
21.: D. J. Wright, The shape of the boundary of Maskit's embedding of the Teichmüller space of punctured tori, preprint, 1990.

Retrieve articles in Conformal Geometry and Dynamics with MSC (2000): 30F40, 30F60

Jouni Parkkonen
Affiliation: Department of Mathematics and Statistics, P.O. Box 35, 40014 University of Jyväskylä, Finland
Email: parkkone@maths.jyu.fi

DOI: 10.1090/S1088-4173-04-00116-X
PII: S 1088-4173(04)00116-X
Received by editor(s): September 10, 2002
Received by editor(s) in revised form: June 17, 2003
Posted: October 14, 2004
Additional Notes: The author was supported by Academy of Finland grants 42948 and 41320
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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