On the Equicontinuity Region of Discrete Subgroups of PU(1,n)

Abstract

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ nℂ preserving the unit ball ℍ nℂ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ nℂ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ nℂ which are tangent to ∂ℍ nℂ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ nℂ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.