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Abstract

A compact Klein surface can be represented in the form D/Γ where D denotes the hyperbolic plane and Γ a non-Euclidean crystallographic (N.E.C.) group of isometries. If Γ + denotes the subgroup of orientation-preserving isometries, then D/Γ + is conformally equivalent to a compact Riemann surface; and if it is hyperelliptic, then D/Γ is called a hyperelliptic Klein surface (H.K.S.). This paper extends the results of the reviewer [same journal Ser. (2) 22 (1971) 117--123; MR0283194 (44 #427)] to characterise H.K.S. and their smooth normal hyperelliptic coverings via N.E.C. groups and their signatures. In addition, the number of hyperelliptic coverings of a given H.K.S. is computed and all results translated into the language of real algebraic curves.