Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T02:02:48.571Z Has data issue: false hasContentIssue false
  • English
  • Français

The limit set of discrete subgroups of PSL(3, ℂ)

Published online by Cambridge University Press:  02 September 2010

WALDEMAR DEL JESÚS BARRERA VARGAS ,
ANGEL CANO CORDERO  and
JUAN PABLO NAVARRETE CARRILLO
WALDEMAR DEL JESÚS BARRERA VARGAS
Affiliation:
Universidad Autónoma de Yucatán, Facultad de Matematicas, Periférico Norte Tablaje Cat 13615 Chuburná Hildalgo, Mérida, Yucatán, México. e-mail: [email protected]
ANGEL CANO CORDERO
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil. e-mail: [email protected]
JUAN PABLO NAVARRETE CARRILLO
Affiliation:
Universidad Autónoma de Yucatán, Facultad de Matematicas, Periférico Norte Tablaje Cat 13615 Chuburná Hildalgo, Mérida, Yucatán, México. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

If Γ is a discrete subgroup of PSL(3, ℂ), it is determined the equicontinuity region Eq(Γ) of the natural action of Γ on ℙ2ℂ. It is also proved that the action restricted to Eq(Γ) is discontinuous, and Eq(Γ) agrees with the discontinuity set in the sense of Kulkarni whenever the limit set of Γ in the sense of Kulkarni, Λ(Γ), contains at least three complex lines in general position. Under some additional hypothesis, it turns out to be the largest open set on which Γ acts discontinuously. Moreover, if Λ(Γ) contains at least four complex lines and Γ acts on ℙ2ℂ without fixed points nor invariant complex lines, then each connected component of Eq(Γ) is a holomorphy domain and a complete Kobayashi hyperbolic space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 1 , January 2011 , pp. 129 - 146
Copyright
Copyright © Cambridge Philosophical Society 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barrera, W. and Navarrete, J. P.Discrete subgroups of PU(2, 1) acting on ℙ2ℂ and the Kobayashi metric. Bull. Braz. Math. Soc. New Series (1) 40 (2009), 99–106.Google Scholar
[2]Cano, A. On discrete subgroups of automorphism of ℙ2ℂ preprint (2009). http://arxiv.org/abs/0806.1336.Google Scholar
[3]Kulkarni, R. S.Groups with domains of discontinuity. Math. Ann. 237 (1978), 253–272.CrossRefGoogle Scholar
[4]Navarrete, J. P.On the limit set of discrete subgroups of PU(2, 1). Geom. Dedicata 122 (2006), 1–13.CrossRefGoogle Scholar
[5]Navarrete, J. P.The trace function and complex Kleinian groups in ℙ2ℂ. Internat. J. Math. 7 (2008), 865–890.CrossRefGoogle Scholar
[6]Patterson, S. J. Lectures on measures on limit sets of Kleinian groups. Analytical Aspects of Hyperbolic Space London Math. Soc. Lecture Note Ser. (Cambridge University Press 1987), 291–323.Google Scholar
[7]Seade, J. and Verjovsky, A.Actions of discrete groups on complex projective spaces. Contemp. Math. 269 (2001), 155–178.CrossRefGoogle Scholar
[8]Seade, J. and Verjovsky, A.Higher dimensional complex Kleinian groups. Math. Ann. 322 (2002), 279–300.CrossRefGoogle Scholar
[9]Sullivan, D.The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Etudes Sci. 50 (1979), 171–202.CrossRefGoogle Scholar