Geodesic
On the topological and metric types of surfaces regularly covering a~closed surface
Izvestiya. Mathematics , Tome 34 (1990) no. 3, pp. 517-553.

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A description is given of the topological types (of which there are six) of noncompact surfaces that can cover a closed surface in a regular fashion. For each of the six topological types, a computation is made of the number of equimorphic types of such surfaces that are equipped with the structure of a Riemannian $2$-manifold regularly covering a closed Riemannian $2$-manifold. Bibliography: 34 titles. 
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R. I. Grigorchuk. On the topological and metric types of surfaces regularly covering a~closed surface. Izvestiya. Mathematics , Tome 34 (1990) no. 3, pp. 517-553. http://geodesic.mathdoc.fr/item/IM2_1990_34_3_a1/

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