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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.19 no.3 Temuco Dec. 2017

http://dx.doi.org/10.4067/S0719-06462017000300069 

Articles

Totally Degenerate Extended Kleinian Groups

Rubén A. Hidalgo1 

1 Universidad de La Frontera, Departamento de Matemática y Estadística, Temuco, Chile. E-mail: ruben.hidalgo@ufrontera.cl

Abstract

The theoretical existence of totally degenerate Kleinian groups is originally due to Bers and Maskit. In fact, Maskit proved that for any co-compact non-triangle Fuchsian group acting on the hyperbolic plane ℍ² there is a totally degenerate Kleinian group algebraically isomorphic to it. In this paper, by making a subtle modification to Maskit’s construction, we show that for any non-Euclidean crystallographic group F, such that ℍ²/F is not homeomorphic to a pant, there exists an extended Kleinian group G which is algebraically isomorphic to F and whose orientation-preserving half is a totally degenerate Kleinian group. Moreover, such an isomorphism is provided by conjugation by an orientation-preserving homeomorphism ϕ : ℍ² → Ω, where Ω is the region of discontinuity of G. In particular, this also provides another proof to Miyachi’s existence of totally degenerate finitely generated Kleinian groups whose limit set contains arcs of Euclidean circles.

Keywords and Phrases: Kleinian Groups; NEC groups

Resumen

La existencia teórica de grupos Kleinianos totalmente degenerados se debe originalmente a Bers y Maskit. De hecho, Maskit demostró que para cualquier grupo Fuchsiano co-compacto y no-triangular actuando en el plano hiperbólico ℍ² existe un grupo Kleiniano totalmente degenerado algebraicamente isomorfo a él. En este artículo, haciendo una modificación sutil a la construcción de Maskit, mostramos que para cualquier grupo cristalográfico no-Euclidiano F tal que ℍ²/F no es homeomorfo a un pantalón, existe un grupo Kleiniano extendido G que es algebraicamente isomorfo a F y cuya mitad que preserva orientación es un grupo Kleiniano totalmente degenerado. Más aún, un tal isomorfismo está dado por la conjugación por un homeomorfismo que preserva orientación ϕ: ℍ² → Ω, donde Ω es la región de discontinuidad de G. En particular, esto también entrega otra demostración del resultado de Miyachi acerca de la existencia de grupos Kleinianos totalmente degenerados finitamente generados cuyo conjunto límite contiene arcos de circunferencias Euclidianas.

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References

[1] L. Bers. On boundaries of Teichmüller spaces and on Kleinian groups: I. Ann. of Math. 91 (1970), 570-600. [ Links ]

[2] V. Chuckrow. On Schottky groups with applications to Kleinian groups. Ann. of Math. 88 (1968), 47-61. [ Links ]

[3] B. Maskit, Kleinian Groups, GMW, Springer-Verlag, 1987. [ Links ]

[4] B. Maskit. On boundaries of Teichm¨uller spaces and on Kleinian groups: II. Ann. of Math. 91 (1970), 607-639. [ Links ]

[5] B. Maskit. On Klein’s Combination Theorem Trans. of the Amer. Math. Soc. 120, No. 3 (1965), 499-509. [ Links ]

[6] B. Maskit. On Klein’s combination theorem III. Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies 66 (1971), Princeton Univ. Press, 297-316. [ Links ]

[7] Maskit, B. On Klein’s combination theorem. IV. Trans. Amer. Math. Soc. 336 (1993), 265-294. [ Links ]

[8] H. Miyachi. Quasi-arcs in the limit set of a singly degenerate group with bounded geometry. In Kleinian Groups and Hyperbolic 3-Manifolds (Eds. Y. Komori, V. Markovic C. Series ) LMS. Lec. Notes 299 (2003), 131-144. [ Links ]

fn1*Partially supported by Project Fondecyt 1150003 and Anillo ACT 1415 PIA-CONICYT

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