The core chain of circles of Maskit's embedding for once-punctured torus groups
Author:
Irene Scorza
Journal:
Conform. Geom. Dyn. 10 (2006), 288-325
MSC (2000):
Primary 30F40; Secondary 57M50
DOI:
https://doi.org/10.1090/S1088-4173-06-00134-2
Published electronically:
October 10, 2006
MathSciNet review:
2261053
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we describe the limit set of a sequence of manifolds
in the boundary of Maskit's embedding of the once-punctured torus. We prove that
contains a chain of tangent circles
that are described from the end invariants of the manifold. In particular, we give estimates in terms of
of the radii
of the circles and prove that
decrease when
tends to infinity. We then apply these results to McShane's identity, to obtain an estimate of the width of the limit set in terms of
.
- 1. H. Akiyoshi, H. Miyachi, and M. Sakuma, A refinement of McShane's identity for quasi-Fuchsian punctured torus groups, Preprint.
- 2.
J. W. Anderson and R. D. Canary, Cores of hyperbolic
-manifolds and limits of Kleinian groups. II, J. London Math. Soc. (2) 61 (2000), no. 2, 489-505. MR 1760675 (2001h:30041)
- 3. A. F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 0698777 (85d:22026)
- 4. L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94-97. MR 0111834 (22:2694)
- 5.
F. Bonahon, Bouts des variétés hyperboliques de dimension
, Ann. of Math. (2) 124 (1986), no. 1, 71-158. MR 0847953 (88c:57013)
- 6. B. Bowditch, The Cannon-Thurston map for punctured-surface groups, Preprint.
- 7. B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3) 77 (1998), no. 3, 697-736. MR 1643429 (99f:57014)
- 8. R. Evans, Strong convergence of sequences of Kleinian groups, Preprint.
- 9. W. J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205-218. MR 0568933 (81e:57002)
- 10. T. Jørgensen, On cyclic groups of Möbius transformations, Math. Scand. 33 (1973), 250-260 (1974). MR 0348103 (50:601)
- 11. -, On pairs of once-punctured tori, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 183-207. MR 2044551 (2005a:30075)
- 12. T. Jørgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), no. 1, 47-72. MR 1060898 (91f:30068)
- 13. L. Keen, B. Maskit, and C. Series, Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets, J. Reine Angew. Math. 436 (1993), 209-219. MR 1207287 (94b:30053)
- 14. L. Keen and C. Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719-749. MR 1241870 (95g:32030)
- 15. B. Maskit, On boundaries of Teichmüller spaces and on Kleinian groups. II, Ann. of Math. (2) 91 (1970), 607-639. MR 0297993 (45:7045)
- 16. -, On boundaries of Teichmüller spaces and on Kleinian groups. II, Ann. of Math. (2) 91 (1970), 607-639. MR 0297993 (45:7045)
- 17. C. T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347 (97f:57022)
- 18. -, Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math. 146 (2001), no. 1, 35-91. MR 1859018 (2004e:30068)
- 19. G. McShane, Simple geodesics and a series constant over Teichmüller space, Invent. Math. 132 (1998), no. 3, 607-632. MR 1625712 (99i:32028)
- 20.
R. Meyerhoff, A lower bound for the volume of hyperbolic
-manifolds, Canad. J. Math. 39 (1987), no. 5, 1038-1056. MR 0918586 (88k:57049)
- 21.
Y. N. Minsky, On rigidity, limit sets, and end invariants of hyperbolic
-manifolds, J. Amer. Math. Soc. 7 (1994), no. 3, 539-588. MR 1257060 (94m:57029)
- 22. -, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559-626. MR 1689341 (2000f:30028)
- 23. H. Miyachi, Moduli of continuity of Cannon-Thurston maps, Spaces of Kleinian Groups (2004), London Math. Soc. Lecture Note Ser., vol. 300, Cambridge Univ. Press, Cambridge, 2004, pp. 1-26.
- 24. J. R. Parker, Tetrahedral decomposition of punctured torus bundles, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 275-291. MR 2044554 (2005g:57037)
- 25. I. Scorza, Fractal curves in the limit sets of simply degenerate once-punctured torus groups, Preprint.
- 26. W. Thurston, Hyperbolic structures on 3-manifolds, II: Surface groups and manifolds which fiber over the circle, Preprint.
- 27. W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 0648524 (83h:57019)
- 28. D. Wright, The shape of the boundary of Maskit's embedding of the Teichmüller space of once-punctured tori, Preprint.
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Additional Information
Irene Scorza
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso, 35 - 16146 Genova, Italy
Email:
scorza@dima.unige.it
DOI:
https://doi.org/10.1090/S1088-4173-06-00134-2
Keywords:
Kleinian groups,
limit sets.
Received by editor(s):
January 19, 2005
Published electronically:
October 10, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.