Totally Degenerate Extended Kleinian Groups

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...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autor principal: Hidalgo,Rubén A.
Lenguaje:English
Publicado: Universidad de La Frontera. Departamento de Matemática y Estadística. 2017
Materias:
Acceso en línea:http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462017000300069
Etiquetas: Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
Descripción
Sumario: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.