Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: (W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p((W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.