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...
Guardado en:
| Autor principal: | |
|---|---|
| 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!
|
| id |
oai:scielo:S0719-06462017000300069 |
|---|---|
| record_format |
dspace |
| spelling |
oai:scielo:S0719-064620170003000692018-06-25Totally Degenerate Extended Kleinian GroupsHidalgo,Rubén A. Kleinian Groups NEC 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 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.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.19 n.3 20172017-12-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462017000300069en10.4067/S0719-06462017000300069 |
| institution |
Scielo Chile |
| collection |
Scielo Chile |
| language |
English |
| topic |
Kleinian Groups NEC groups |
| spellingShingle |
Kleinian Groups NEC groups Hidalgo,Rubén A. Totally Degenerate Extended Kleinian Groups |
| description |
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. |
| author |
Hidalgo,Rubén A. |
| author_facet |
Hidalgo,Rubén A. |
| author_sort |
Hidalgo,Rubén A. |
| title |
Totally Degenerate Extended Kleinian Groups |
| title_short |
Totally Degenerate Extended Kleinian Groups |
| title_full |
Totally Degenerate Extended Kleinian Groups |
| title_fullStr |
Totally Degenerate Extended Kleinian Groups |
| title_full_unstemmed |
Totally Degenerate Extended Kleinian Groups |
| title_sort |
totally degenerate extended kleinian groups |
| publisher |
Universidad de La Frontera. Departamento de Matemática y Estadística. |
| publishDate |
2017 |
| url |
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462017000300069 |
| work_keys_str_mv |
AT hidalgorubena totallydegenerateextendedkleiniangroups |
| _version_ |
1714206798255226880 |