Geodesic
The Classification of Non-Euclidean Plane Crystallographic Groups
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 1192-1205

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This paper deals with the algebraic classification of non-euclidean plane crystallographic groups (NEC groups, for short) with compact quotient space. The groups considered are the discrete groups of motions of the Lobatschewsky or hyperbolic plane, including those which contain orientation-reversing reflections and glide-reflections. The corresponding problem for Fuchsian groups, which contain only orientable transformations, is essentially solved in the work of Fricke and Klein (6). 
Macbeath, A. M. The Classification of Non-Euclidean Plane Crystallographic Groups. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 1192-1205. doi: 10.4153/CJM-1967-108-5
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