Schlicht envelopes of holomorphy and topology

Abstract

Let Ω be a domain in ${\mathbb{C}^{2}}$, and let ${\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}}$ be its envelope of holomorphy. Also let ${\Omega'=\pi(\tilde{\Omega})}$ with ${i: \Omega \hookrightarrow \Omega'}$ the inclusion. We prove the following: if the induced map on fundamental groups ${i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')}$ 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.