On Limit Sets of 4-dimensional Kleinian Groups with 3 Generators
Keita SAKUGAWA
Source: Tokyo J. of Math. Volume 33, Number 1
(2010), 165-182.
Abstract
In this paper, we consider a quaternionic representation of a 4-dimensional Kleinian group $G$ with 3 generators $f,g$ and $h$, where $g$ and $h$ are simple parabolic, $[g,h]= id$, and $[f,g],[f,h]$ are order-2 elliptic elements. We parameterize such $f,g$ and $h$ up to conjugacy and we simulate the shape of the limit set $\Lambda(G)$ using computer.
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Permanent link to this document: http://projecteuclid.org/euclid.tjm/1279719585
Digital Object Identifier: doi:10.3836/tjm/1279719585
Zentralblatt MATH identifier: 05776106
Mathematical Reviews number (MathSciNet): MR2682888
References
Ahara K., Sphairahedra and 4-dimensional quasi-fuchsian groups (in Japanese), RIMS proceedings, 1329 (2003), 109--114.
Ahara K. and Araki Y., Spheirahedral approach to parameterize visible three dimensional quasi-Fuchsian fractals, Proc. of the CGI (2003), 226--229.
Akiyoshi H., Sakuma M., Wada M. and Yamashita Y., Jorgensen's picture of quasifuchsian punctured torus groups, Kleinian groups and hyperbolic 3-manifolds, London Math. Soc., Lecture Note Series. 299 (2003), 247--273.
Akiyoshi H. and Sakuma M., Comparing two convex hull constructions of cusped hyperbolic manifolds, Kleinian groups and hyperbolic 3-manifolds, London Math. Soc., Lecture Note Series. 299 (2003), 209--246.
Mathematical Reviews (MathSciNet): MR2044552
Zentralblatt MATH: 1048.57007
Digital Object Identifier: doi:10.1017/CBO9780511542817.011
Akiyoshi H., Miyachi H. and Sakuma M., A refinement of McShane's identity for quasifuchsian punctured torus groups, Kleinian groups and hyperbolic 3-manifolds, London Math. Soc., Lecture Note Series. 329 (2006), 151--185.
Araki Y. and Ito K., An extension of the Maskit slice for 4-dimensional Kleinian groups, Conform. Geom. Dyn. 12 (2008), 199--226.
Mathematical Reviews (MathSciNet): MR2466017
Digital Object Identifier: doi:10.1090/S1088-4173-08-00187-2
Araki Y., Ito K. and Komori Y., A note on a 3-dimensional extension of the Maskit slice (in Japanese), RIMS Proceedings, 1571 (2007), 172--192.
Cao W., Parker J. and Wang X., On the classification of quaternionic Möbius transformations, Math. Proc. Cambridge Philosophical Soc., 137:2 (2004), 349--362.
Mathematical Reviews (MathSciNet): MR2092064
Digital Object Identifier: doi:10.1017/S0305004104007868
Epstein D. B. A., Marden A. and Markovic V., Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math., 159 (2004), 305--336.
Mathematical Reviews (MathSciNet): MR2052356
Zentralblatt MATH: 1064.30044
Digital Object Identifier: doi:10.4007/annals.2004.159.305
JSTOR: links.jstor.org
Hersonsky S., A generalization of the Shimizu-Leutbecher and Jørgensen inequalities to Möbius transformations in $\textbf{R}^{\textbf{N}}$, Proc. of the American Math. Soc., 121:1 (1994), 209--215.
Mathematical Reviews (MathSciNet): MR1182701
JSTOR: links.jstor.org
Hertrich-Jeromin U., Introduction to Möbius Differential Geometry, London Math. Soc., Lecture Note Series. 300 (2003).
Mathematical Reviews (MathSciNet): MR2004958
Hidalgo R. and Maskit B., On neoclassical schottky groups, Trans. of the American Math. Soc., 358:11 (2006), 4765--4792.
Mathematical Reviews (MathSciNet): MR2231871
Zentralblatt MATH: 1125.30032
Digital Object Identifier: doi:10.1090/S0002-9947-05-03792-X
Kapovich M., Hyperbolic Manifolds and Discrete Groups, Birkhäuser, 2001.
Mathematical Reviews (MathSciNet): MR1792613
Kapovich M., Topological aspects of Kleinian groups in several dimensions, Proc. of 3-rd Ahlfors-Bers Colloquium, (2002).
Kido T., Möbius transformations on quaternion, Tohoku Math. J., to, appear.
Komori Y., Sugawa T., Wada M. and Yamashita Y., Drawing Bers Embeddings of the Teichmuller Space of Once-Punctured Tori, Exper. Math., 15 (2006), 51--60.
Mathematical Reviews (MathSciNet): MR2229385
Project Euclid: euclid.em/1150476903
Lanphier D. and Rosenhouse J., A decomposition theorem for Cayley graph of Picard group quotients, The Journal of Combinatorial Math. and Combinatorial Computing, 50 (2004), 95--104.
Mathematical Reviews (MathSciNet): MR2075858
Zentralblatt MATH: 1055.05079
Maskit B., Kleinian Groups, Springer (1988).
Mathematical Reviews (MathSciNet): MR959135
Miyachi H., Private introduction of 3-dimensional hyperbolic geometry, preprint.
Mumford D., Series C. and Wright D., Indra's Pearls, Cambridge Press, 2003.
POV-Ray, http://www.povray.org/
Sakugawa K., ``Norio'', software.
Taniguchi M. and Matsuzaki K., Hyperbolic manifolds and Kleinian groups (in Japanese), Nihon-Hyoronsha, 1993.
Taniguchi M. and Okumura Y., Introduction of hyperbolic geometry (in Japanese), Baifu-kan, 1996.
Wada M., OPTi's Algorithm for Discreteness Determination, Exper. Math., 15 (2006), 61--66.
Mathematical Reviews (MathSciNet): MR2229386
Zentralblatt MATH: 1107.30033
Project Euclid: euclid.em/1150476904
Wada M., OPTi's Algorithm for Drawing the Limit Set, Interdisciplinary Information Sciences, 9 (2003), 183--187.
Mathematical Reviews (MathSciNet): MR2023117
Zentralblatt MATH: 1079.57501
Digital Object Identifier: doi:10.4036/iis.2003.183
Wright D., Searching for the cusp, Spaces of Kleinian Groups, Cambridge University Press Lond. Math. Soc. Lec. Notes 329 (2005), 301--336.
Mathematical Reviews (MathSciNet): MR2258756
Zentralblatt MATH: 1103.30029
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Tokyo Journal of Mathematics
