Free groups generated by two screw parabolic maps

In this paper, we provide sufficient conditions for groups generated by two screw parabolic maps of \({\textrm{PU}}(2,1)\) $\text{PU}(2,1)$ with distinct fixed points to be discrete and free. As an example, we apply our results to the classical group \(\langle g_\theta , f_\theta \rangle \) $⟨g_{\theta },f_{\theta }⟩$ constructed in Gusevskii, N., Parker, J.R.: Representations of free Fuchsian groups in complex hyperbolic space. Topology 39, 33–60 (2000).