Real forms of a Riemann surface of even genus

Authors:
G. Gromadzki and M. Izquierdo

Journal:
Proc. Amer. Math. Soc. **126** (1998), 3475-3479

MSC (1991):
Primary 20F10, 30F10; Secondary 30F35, 51M10, 14H99

DOI:
https://doi.org/10.1090/S0002-9939-98-04735-2

MathSciNet review:
1485478

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Abstract: Natanzon proved that a Riemann surface of genus has at most conjugacy classes of symmetries, and this bound is attained for infinitely many genera . The aim of this note is to prove that a Riemann surface of even genus has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus has at most four real forms which are not birationally equivalent.

**1.**N. L. Alling, N. Greenleaf,*Foundations of the theory of Klein surfaces*, Lecture Notes in Math., vol.**219**, Springer-Verlag (1971). MR**88m:26027****2.**E. Bujalance, J. J. Etayo, J. M. Gamboa, and G. Gromadzki,*A combinatorial approach to groups of automorphisms of bordered Klein surfaces*, Lecture Notes in Math., vol.**1439**, Springer Verlag (1990). MR**92a:14018****3.**E. Bujalance, G. Gromadzki, and D. Singerman,*On the number of real curves associated to a complex algebraic curve*, Proc. Amer. Math. Soc.**120**(2) (1994), 507-513. MR**94d:20054****4.**S. M. Natanzon,*On the order of a finite group of homeomorphisms of a surface into itself and the real number of real forms of a complex algebraic curve*, Dokl. Akad. Nauk SSSR**242**(1978), 765-768. (Soviet Math. Dokl.**19**(5), (1978), 1195-1199.) MR**82b:14019****5.**D. Singerman,*Symmetries and pseudo-symmetries of hyperelliptic surfaces*, Glasgow Math. J.**31**(1980), 39-49. MR**81c:30080****6.**D. Singerman,*Finitely maximal Fuchsian groups*, J. London Math. Soc.**6**(1972), 29-38. MR**48:529**

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Additional Information

**G. Gromadzki**

Affiliation:
Institute of Mathematics University of Gdańsk, ul. Wita Stowsza 57, 80-952 Gdańsk, Poland

**M. Izquierdo**

Affiliation:
Department of Mathematics, Mälardalen University, 721 23 Västerås, Sweden

Email:
mio@mdh.se

DOI:
https://doi.org/10.1090/S0002-9939-98-04735-2

Received by editor(s):
April 14, 1997

Additional Notes:
The second author was partially supported by The Swedish Natural Science Research Council (NFR)

Communicated by:
Linda Keen

Article copyright:
© Copyright 1998
American Mathematical Society