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A combinatorial approach to the symmetries of $$M$$ and $$M-1$$ Riemann surfaces. (English) Zbl 0788.30036
Discrete groups and geometry, Proc. Conf., Birmingham/UK 1991, Lond. Math. Soc. Lect. Note Ser. 173, 16-25 (1992).
[For the entire collection see Zbl 0746.00069.]
A compact Riemann surface $$X$$ of genus $$g$$ is said to be an $$M$$ (respectively $$M-1)$$ Riemann surface if it admits a symmetry with $$g+1$$ fixed curves (respectively $$g$$ fixed curves). S. M. Natanzon proved some topological nature of the symmetries of $$M$$ and $$M-1$$ Riemann surfaces and, in the hyperelliptic case, the classification of such symmetries up to conjugation in the automorphism group. In this paper, the author gives a new proof of the results of S. M. Natanzon and some improvements of Natanzon’s work in the non-hyperelliptic case, by using the combinatorial theory of non-Euclidean crystallographic groups.
Reviewer: Li Zhong (Beijing)

##### MSC:
 30F50 Klein surfaces 30F10 Compact Riemann surfaces and uniformization