On the Limit Set of Discrete Subgroups of PU(2,1)

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Let G be a discrete subgroup of PU(2,1); G acts on \(P^2_\mathbb C\) preserving the unit ball \(\mathbf H^2 _{\mathbb C}\), equipped with the Bergman metric. Let \(L(G) \subset S^3 = \partial \mathbf H^2 _{\mathbb C}\) be the limit set of G in the sense of Chen–Greenberg, and let\(\Lambda(G) \subset P^2_{\mathbb C}\) be the limit set of the G-action on\(P^2_{\mathbb C}\) in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S 3 and Λ(G) is the union of all complex projective lines in \(P^2_\mathbb C\) which are tangent to S 3 at a point in L(G).