Totally Degenerate Extended Kleinian Groups

  • Rubén A. Hidalgo Departamento de Matemáticas y Estadística, Facultad de Ingeniería y Ciencias, Universidad de La Frontera, Casilla 54-D, Temuco - Chile.

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.

Published
2017-10-01
How to Cite
HIDALGO, Rubén A.. Totally Degenerate Extended Kleinian Groups. CUBO, A Mathematical Journal, [S.l.], v. 19, n. 3, p. 69–77, oct. 2017. ISSN 0719-0646. Available at: <http://revistas.ufro.cl/index.php/cubo/article/view/1872>. Date accessed: 17 oct. 2018.