Covering properties of open continuous mappings having two valences between Riemann surfaces. (English) Zbl 0856.30005

Lyzzaik, Abdallah

Math. Proc. Camb. Philos. Soc. 118, No. 2, 321-340 (1995).

Let \(X\) denote an open Riemann surface with finite genus and finite number of boundary components, and let \(Y\) denote a closed Riemann surface. An open continuous function from \(X\) to \(Y\) is said to be a \((p,q)\)-map, \(0 < q < p\), if it has a finite number of branch points

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and assumes every point in \(Y\) either \(p\) or \(q\) times, counting multiplicity, with at most a finite number of exceptions. The study of such maps was initiated, in a special case, by the reviewer and W. E. Kirwan [J. Lond. Math. Soc., II. Ser. 19, 93-101 (1979; Zbl 0399.30007)]; this paper generalises more recent work by U. Srebro and B. Wajnryb [J. Anal. Math. 46, 283-303 (1986; Zbl 0604.30012)] and A. K. Lyzzaik and K. Stephenson [Trans. Am. Math. Soc. 327, No. 2, 525-566 (1991; Zbl 0741.57001)]. The author proves here that, if \(f : X \to Y\) is a \((p,q)\)-map, then there exists a \(p\)-fold covering surface \((\widetilde Y, \pi)\) of \(Y\) and an embedding \(\varphi : X \to \widetilde Y\) such that \(f \equiv \pi \circ \varphi\). (It follows that one can view the image surface of a \((p,q)\)-map as a subset of a \(p\)-fold covering of the image surface.) The proof uses the idea of tacking on appropriate additional branch points to the image surface of the \((p,q)\)-map. The author also shows that the hypothesis of \((p,q)\)-maps cannot be weakened for the existence result to hold.

Reviewer: D.A.Brannan (Milton Keynes)

MSC:

30C20

Conformal mappings of special domains

Keywords:

covering surface; embedding theorems; open continuous mappings

Citations:

Zbl 0399.30007; Zbl 0604.30012; Zbl 0741.57001

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