Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms

  • Ewa Kozłowska-Walania

Abstract

We study the symmetric Riemann surfaces for which the group of orientation preserving automorphisms acts without fixed points. We show that any finite group can give rise to such an action, determine the maximal number of non-conjugate symmetries for such surfaces and find a sharp upper bound on maximal total number of ovals for a set of $k$ symmetries with ovals. We also solve the minimal genus problem for dihedral groups acting on the surfaces described above, for odd genera.

References

Bujalance, E., Normal N.E.C. signatures, Illinois J. Math. 26 (1982), no. 3, 519–530.

Bujalance, E. and Costa, A. F., On symmetries of $p$-hyperelliptic Riemann surfaces, Math. Ann. 308 (1997), no. 1, 31–45. https://doi.org/10.1007/s002080050062

Bujalance, E. and Costa, A. F., On the group generated by three and four anticonformal involutions of Riemann surfaces with maximal number of fixed curves, in “Mathematical contributions in honor of Professor Enrique Outerelo Dom\'ınguez”, Homen. Univ. Complut., Editorial Complutense, Madrid, 2004, pp. 73–76.

Bujalance, E., Costa, A. F., and Singerman, D., Application of Hoare's theorem to symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 307–322.

Bujalance, E., Gromadzki, G., and Izquierdo, M., On real forms of a complex algebraic curve, J. Aust. Math. Soc. 70 (2001), no. 1, 134–142. https://doi.org/10.1017/S1446788700002329

Bujalance, E. and Singerman, D., The symmetry type of a Riemann surface, Proc. London Math. Soc. (3) 51 (1985), no. 3, 501–519. https://doi.org/10.1112/plms/s3-51.3.501

Estévez, J. L. and Izquierdo, M., Non-normal pairs of non-Euclidean crystallographic groups, Bull. London Math. Soc. 38 (2006), no. 1, 113–123. https://doi.org/10.1112/S0024609305017984

Greenberg, L., Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. https://doi.org/10.1090/S0002-9904-1963-11001-0

Gromadzki, G., On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces, J. Pure Appl. Algebra 121 (1997), no. 3, 253–269. https://doi.org/10.1016/S0022-4049(96)00068-0

Gromadzki, G., On ovals on Riemann surfaces, Rev. Mat. Iberoamericana 16 (2000), no. 3, 515–527. https://doi.org/10.4171/RMI/282

Gromadzki, G. and Kozłowska-Walania, E., On fixed points of doubly symmetric Riemann surfaces, Glasg. Math. J. 50 (2008), no. 3, 371–378. https://doi.org/10.1017/S0017089508004278

Gromadzki, G. and Kozłowska-Walania, E., On ovals of non-conjugate symmetries of Riemann surfaces, Internat. J. Math. 20 (2009), no. 1, 1–13. https://doi.org/10.1142/S0129167X09005145

Izquierdo, M. and Singerman, D., Pairs of symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 3–24.

Kozłowska-Walania, E., On commutativity and ovals for a pair of symmetries of a Riemann surface, Colloq. Math. 109 (2007), no. 1, 61–69. https://doi.org/10.4064/cm109-1-5

Kozłowska-Walania, E., On $p$-hyperellipticity of doubly symmetric Riemann surfaces, Publ. Mat. 51 (2007), no. 2, 291–307. https://doi.org/10.5565/PUBLMAT_51207_02

Kozłowska-Walania, E., Non-commuting pairs of symmetries of Riemann surfaces, Rocky Mountain J. Math. 43 (2013), no. 3, 989–1014. https://doi.org/10.1216/RMJ-2013-43-3-989

Melekoglu, A., Symmetries of Riemann surfaces and regular maps, Ph.D. thesis, Southampton, United Kingdom, 1998.

Natanzon, S. M., Finite groups of homeomorphisms of surfaces, and real forms of complex algebraic curves, Trudy Moskov. Mat. Obshch. 51 (1988), 3–53, translation Trans. Moscow Math. Soc. 1989, 1–51.

Singerman, D., On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Philos. Soc. 76 (1974), 233–240. https://doi.org/10.1017/s0305004100048891
Published
2020-09-03
How to Cite
Kozłowska-Walania, E. (2020). Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms. MATHEMATICA SCANDINAVICA, 126(3), 479-492. https://doi.org/10.7146/math.scand.a-121167
Section
Articles