A combinatorial approach to the symmetries of \(M\) and \(M-1\) Riemann surfaces. (English) Zbl 0788.30036

Bujalance, E.; Costa, A. F.

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

Keywords:

hyperelliptic case; combinatorial theory of non-Euclidean crystallographic groups