Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

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

WALDEMAR DEL JESÚS BARRERA VARGASa1, ANGEL CANO CORDEROa2 and JUAN PABLO NAVARRETE CARRILLOa3

a1 Universidad Autónoma de Yucatán, Facultad de Matematicas, Periférico Norte Tablaje Cat 13615 Chuburná Hildalgo, Mérida, Yucatán, México. e-mail: bvargas@uady.mx

a2 Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil. e-mail: acano@impa.br

a3 Universidad Autónoma de Yucatán, Facultad de Matematicas, Periférico Norte Tablaje Cat 13615 Chuburná Hildalgo, Mérida, Yucatán, México. e-mail: jp.navarrete@uady.mx

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 ℙ2. 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 ℙ2 without fixed points nor invariant complex lines, then each connected component of Eq(Γ) is a holomorphy domain and a complete Kobayashi hyperbolic space.

(Received January 11 2010)

(Online publication September 02 2010)