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The holomorphic universal covers of spaces of polynomials without multiple roots. (English) Zbl 0789.32012
Let \(G_ n\) be the manifold of polynomials of degree \(n\) in one complex variable with leading coefficient 1 and without multiple roots. \(G_ n\) is covered by the manifold \(E_ n=\{z \in \mathbb{C}^ n:z_ i \neq z_ j\}\). The author studies the universal cover of \(E_ n\). It is isomorphic to \(\mathbb{C}^ 2\times\widetilde M_{n-2}\), where \(M_ n=\{z \in \mathbb{C}^ n :z_ i \neq 0,1;z_ i \neq z_ j\}\). The author uses quasiconformal mappings to show that the universal cover \(\widetilde M_ n\) of \(M_ n\) is a Teichmüller space. It is shown that \(SE_ n=\{z \in E_ n: \Sigma z_ j=0, \prod_{i<j}(z_ i-z_ j)=1\}\) covers \(M_{n-2}\) and that holomorphic functions on \(SE_ n\) which omit the values 0 and 1 descend to holomorphic functions on \(M_{n-2}\). The author also gives a short proof that \(M_ n\), and therefore \(\widetilde M_ n\), is complete hyperbolic in the sense of Kobayashi, and poses some questions about the holomorphic contractibility of \(\widetilde M_ n\). The necessary material on quasiconformal mappings and Teichmüller spaces is reviewed at the beginning of the paper.

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F60 Teichmüller theory for Riemann surfaces