A note on the construction of a certain class of Kleinian groups
Author:
Ricardo Bianconi
Journal:
Proc. Amer. Math. Soc. 123 (1995), 3119-3124
MSC:
Primary 30F40; Secondary 20H10
DOI:
https://doi.org/10.1090/S0002-9939-1995-1277097-0
MathSciNet review:
1277097
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove that if $\{ {S_1},{S_1}’, \ldots ,{S_n},S_n’\}$ is a collection of distinct spheres in ${\mathbb {R}^m}$ with common exterior, and ${g_1}, \ldots ,{g_n}$ are Möbius transformations such that for each i, ${S_i}$ is the isometric sphere of ${g_i}$ and $S_i’$ is the isometric sphere of $g_i^{ - 1}$ and such that ${g_i}$ maps points of contact of ${S_i}$, to points of contact of $S_i’$, then the group G generated by the ${g_i}$’s is Kleinian.
- M. Bestvina and D. Cooper, A wild Cantor set as the limit set of a conformal group action on $S^3$, Proc. Amer. Math. Soc. 99 (1987), no. 4, 623–626. MR 877028, DOI https://doi.org/10.1090/S0002-9939-1987-0877028-4 R. Bianconi, N. Gusevskii, and H. Klimenko, Schottky-type groups and Kleinian groups acting on ${\mathbb {R}^3}$, preprint, 1994.
- Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
- Bernard Maskit, On Klein’s combination theorem, Trans. Amer. Math. Soc. 120 (1965), 499–509. MR 192047, DOI https://doi.org/10.1090/S0002-9947-1965-0192047-1
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Additional Information
Keywords:
Kleinian groups,
parabolic,
infinite cycle transformations,
isometric spheres,
Poincaré’s Polyhedron Theorem
Article copyright:
© Copyright 1995
American Mathematical Society
