## Tokyo Journal of Mathematics

### On Limit Sets of 4-dimensional Kleinian Groups with 3 Generators

Keita SAKUGAWA

#### 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.

#### Article information

Source
Tokyo J. of Math. Volume 33, Number 1 (2010), 165-182.

Dates
First available in Project Euclid: 21 July 2010

Permanent link to this document
http://projecteuclid.org/euclid.tjm/1279719585

Digital Object Identifier
doi:10.3836/tjm/1279719585

Mathematical Reviews number (MathSciNet)
MR2682888

Zentralblatt MATH identifier
1201.30053

#### Citation

SAKUGAWA, Keita. On Limit Sets of 4-dimensional Kleinian Groups with 3 Generators. Tokyo J. of Math. 33 (2010), no. 1, 165--182. doi:10.3836/tjm/1279719585. http://projecteuclid.org/euclid.tjm/1279719585.

#### 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.
• 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.
• 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.
• Epstein D. B. A., Marden A. and Markovic V., Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math., 159 (2004), 305–336.
• 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.
• Hertrich-Jeromin U., Introduction to Möbius Differential Geometry, London Math. Soc., Lecture Note Series. 300 (2003).
• Hidalgo R. and Maskit B., On neoclassical schottky groups, Trans. of the American Math. Soc., 358:11 (2006), 4765–4792.
• Kapovich M., Hyperbolic Manifolds and Discrete Groups, Birkhäuser, 2001.
• 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.
• 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.
• Maskit B., Kleinian Groups, Springer (1988).
• 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.
• Wada M., OPTi's Algorithm for Drawing the Limit Set, Interdisciplinary Information Sciences, 9 (2003), 183–187.
• Wright D., Searching for the cusp, Spaces of Kleinian Groups, Cambridge University Press Lond. Math. Soc. Lec. Notes 329 (2005), 301–336.