Some special Kleinian groups and their orbifolds

Authors

  • Rubén A. Hidalgo Universidad Técnica Federico Santa María.

DOI:

https://doi.org/10.4067/S0716-09172002000100003

Keywords:

Kleinian groups, isometries, hyperbolic spaces, grupos kleinianos, isometrías, espacios hiperbólicos.

Abstract

Let us consider an abstract group with the following presentation

where ki, lj ? {2,..., ?}. We provide conditions in order to find a faithful, discrete and geometrically finite representation ? : G? PSL(2; C), that is, to represent G as a group of isometries of the hyperbolic three space H³.


Author Biography

Rubén A. Hidalgo, Universidad Técnica Federico Santa María.

Departamento de Matemática.

References

[1] H.S.M. Coxeter and W.O.J. Moser. Generators and Relations for Discrete Groups. Springer-Verlag, (1980).

[2] M. Hagelberg. Generalized triangle groups and 3-dimensional orbifolds. Amer. Math. Soc. Contemp. Math. 184, pp. 185-192, (1995).

[3] M. Hagelberg, M. MacClaughlan and G. Rosenberger. On discrete generalized triangle groups. Proc Edinburg Math. Soc., II 38, pp. 397-412, (1995).

[4] M. Hagelberg and A.Y. Vesnin. On a family of hyperbolic dihedral ?(p=q)-orbifolds. Vychisl. Sist. 155, pp. 15-36 (Russian), (1996).

[5] H. Helling, J. Mennicke and E.B. Vinberg. On some general triangle groups and 3¡dimensional orbifolds. Trans. Mosc. Math. Soc., pp. 1-21, (1995).

[6] M. Hagelberg and R. Hidalgo. Generalized Coxeter groups and their orbifolds. Revista Matem´atica Iberoamericana 13 (1997), pp. 543-566, (1997).

[7] B. Maskit. Kleinian groups. Springer-Verlag, Berlin and New York, (1972).

[8] J.G. Ractliffe. Foundations of Hyperbolic Manifolds. Graduate Texts in Math., Springer-Verlag, (1994).

[9] W. Scott. The geometries of three manifolds. Bull. London Math. Soc. 15, pp. 407-487, (1983).

[10] W.P. Thurston. The geometry and topology of 3¡manifolds. Lecture Notes, Princeton Univ.., (1980).

[11] S.V. Tsaranov. On a generalization of Coxeter groups. Alg. Groups Geom. 6, 281-318 (1989).

[12] . Finite generalize Coxeter groups. Alg. Groups Geom. 6, pp. 421-452, (1989).

Published

2017-05-22

How to Cite

[1]
R. A. Hidalgo, “Some special Kleinian groups and their orbifolds”, Proyecciones (Antofagasta, On line), vol. 21, no. 1, pp. 21-50, May 2017.

Issue

Section

Artículos