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On topological and metric types of surfaces regularly covering a closed surface. (Russian) Zbl 0686.57001

Call an oriented noncompact surface L a \(\Gamma\)-surface if it admits a properly discontinuous action of a group \(\Gamma\) of homeomorphisms such that L/\(\Gamma\) is compact. The author solves the problem of topological classification of \(\Gamma\)-surfaces and proves that there exist six types of them: a punctured sphere, a double punctured sphere, a sphere without a Cantor set, an oriented suface with one nonplanar end, an oriented surface with two nonplanar ends and an oriented surface with a Cantor set of nonplanar ends. Then he obtains the same type of results for surfaces which are regular coverings of oriented compact surfaces, genus \(>1\), and for nonoriented cases. The author also estimates the number of quasiconformal and equimorphic (a homeomorphism is called equimorphism if it is equicontinuous in both sides) classes of \(\Gamma\)-surfaces and regular coverings in each class of topologically equivalent surfaces, and gives some examples.
Reviewer: V.B.Marenich

MSC:

57M10 Covering spaces and low-dimensional topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57S30 Discontinuous groups of transformations
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