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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.

##### MSC:
 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