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

Abstract

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).