Opuscula Math. 36, no. 1 (2016), 103-122
http://dx.doi.org/10.7494/OpMath.2016.36.1.103

 
Opuscula Mathematica

On triangular (Dn)-actions on cyclic p-gonal Riemann surfaces

Ewa Tyszkowska

Abstract. A compact Riemann surface \(X\) of genus \(g\gt 1\) which has a conformal automorphism \(\rho\) of prime order \(p\) such that the orbit space \(X/ \langle \rho \rangle \) is the Riemann sphere is called cyclic \(p\)-gonal. Exceptional points in the moduli space \(\mathcal{M}_g\) of compact Riemann surfaces of genus \(g\) are unique surface classes whose full group of conformal automorphisms acts with a triangular signature. We study symmetries of exceptional points in the cyclic \(p\)-gonal locus in \(\mathcal{M}_g\) for which \(\text{Aut}(X)/ \langle \rho \rangle\) is a dihedral group \(D_n\).

Keywords: Riemann surface, symmetry, triangle group, Fuchsian group, NEC group.

Mathematics Subject Classification: 30F10, 14H37.

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  1. R.D.M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398-408.
  2. R.D.M. Accola, On cyclic trigonal Riemann surfaces, Trans. Amer. Math. Soc. 283 (1984) 2, 423-449.
  3. G.V. Belyi, On Galois extensions of maximal cyclotomic field, Math. USSR Izvestiya 14 (1980), 247-256.
  4. S.A. Broughton, E. Bujalance, A.F. Costa, J.M. Gamboa, G. Gromadzki, Symmetries of Accola-Maclachlan and Kulkarni Surfaces, Proc. Amer. Math. Soc. 127 (1999), 637-646.
  5. E. Bujalance, F.J. Cirre, J.M. Gamboa, G. Gromadzki, Symmetry Types of Hyperelliptic Riemann Surfaces, Mémoires de la Société Mathématique De France (2001).
  6. G. Castelnuevo, Sulle serie algebriche di gruppi di punti appartenenti ad una curve algebraica, Rend. R. Academia Lincei Ser. 5 XV (1906), (Memorie scelte p. 509).
  7. H.M. Farkas, I. Kra, Riemann Surfaces, Graduate Text in Mathematics, Springer-Verlag, New York, 1980.
  8. G. Gromadzki, On a Harnack-Natanzon Theorem for the family of real forms of Riemann surfaces, J. Pure Appl. Algebra 121 (1997), 253-269.
  9. A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Kurven, Math. Ann. 10 (1876), 189-199.
  10. A. Hurwitz, Über alebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1983).
  11. G.A. Jones, D. Singerman, P.D. Watson, Symmetries of quasiplatonic Riemann surfaces, Rev. Mat. Iberoam. (to appear).
  12. A.M. Macbeath, The classification of non-Euclidean plane crystallographic groups, Canad. J. Math. 19 (1966), 1192-1205.
  13. A.M. Macbeath, Action of automorphisms of a compact Riemann surface on the first homology group, Bull. Lond. Math. Soc. 5 (1973), 103-108.
  14. A.M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90-96.
  15. C. Maclachlan, Smooth coverings of hyperelliptic surfaces, Quart. Journal of Math. Oxford 2 (1971), 117-123.
  16. D. Singerman, Symmetries of Riemann surfaces with large automorphism group, Math. Ann. 210 (1974), 17-32.
  17. D. Singerman, Non-Euclidean crystallographic groups and Riemann surfaces, Ph.D. Thesis, Univ. of Birmingham, 1969.
  18. D. Singerman, Finitely maximal Fuchsian groups, J. Lond. Math. Soc. (2) 6 (1972), 29-38.
  19. D. Singerman, The remarkable Accola-Maclachlan surfaces, [in:] Contemp. Math., vol. 629, Amer. Math. Soc., Providence, RI, 2014, pp. 315-322.
  20. E. Tyszkowska, Topological classification of conformal actions on cyclic \(p\)-gonal Riemann surfaces, Journal of Algebra 344 (2011), 296-312.
  21. E. Tyszkowska, A. Weaver, Exceptional points in the elliptic-hyperelliptic locus, J. Pure Appl. Algebra 212 (2008), 1415-1426.
  22. A. Weaver, Hyperelliptic surfaces and their moduli, Geom. Dedicata 103 (2004), 69-87.
  23. H.C. Wilkie, On non-Euclidean crystallographic groups, Math. Z. 91 (1966), 87-102.
  • Ewa Tyszkowska
  • University of Gdańsk, Institute of Mathematics, Wita Stwosza 57, 80-952 Gdańsk, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2014-11-11.
  • Revised: 2015-02-16.
  • Accepted: 2015-02-17.
  • Published online: 2015-09-19.
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Cite this article as:
Ewa Tyszkowska, On triangular (Dn)-actions on cyclic p-gonal Riemann surfaces, Opuscula Math. 36, no. 1 (2016), 103-122, http://dx.doi.org/10.7494/OpMath.2016.36.1.103

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