Discrete subgroups of \(\text{PU}(2, 1)\) acting on \(P_{\mathbb C}^2\) and the Kobayashi metric (Q2271923)

\(\text{PU}(2,1)\) is the group of isometries of the complex hyperbolic space \(\mathcal{H}_{\mathbb C}^2=\{[z_1:z_2:z_3]\in P_{\mathbb C}^2: |z_1|^2+|z_2|^2-|z_3|^2>0\}\), equipped with the Bergman metric. Let \(G\) be an infinite, discrete subgroup of \(\text{PU}(2,1)\), acting on \(P_{\mathbb C}^2\) without invariant lines, and let \(\Omega(G)\subset P_{\mathbb C}^2\) be its domain of discontinuity according to \textit{R. S.~Kulkarni} [Math. Ann. 237, 253--272 (1978; Zbl 0369.20028)]. It is shown that the component of \(\Omega(G)\) containing \(\mathcal H_{\mathbb C}^2\) is \(G\)-invariant and complete Kobayashi hyperbolic.