Grigorchuk, R. I. On topological and metric types of surfaces regularly covering a closed surface. (Russian) Zbl 0686.57001 Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 3, 498-536 (1989). 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 Cited in 1 Review MSC: 57M10 Covering spaces and low-dimensional topology 57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) 57S30 Discontinuous groups of transformations Keywords:noncompact surface; \(\Gamma\)-surface; properly discontinuous action; topological classification of \(\Gamma\)-surfaces; punctured sphere; double punctured sphere; sphere without a Cantor set; nonplanar ends; regular coverings PDFBibTeX XMLCite \textit{R. I. Grigorchuk}, Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 3, 498--536 (1989; Zbl 0686.57001)