Symmetries of Accola-Maclachlan and Kulkarni surfaces

Broughton, S.; Bujalance, E.; Costa, A.; Gamboa, J.; Gromadzki, G.

doi:10.1090/S0002-9939-99-04534-7

Symmetries of Accola-Maclachlan and Kulkarni surfaces
HTML articles powered by AMS MathViewer

by S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa and G. Gromadzki PDF

Proc. Amer. Math. Soc. 127 (1999), 637-646 Request permission

Abstract:

For all $g \ge 2$ there is a Riemann surface of genus $g$ whose automorphism group has order $8g+8$, establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large $g$ the Accola-Maclachlan surface was unique for $g= 0,1,2\mod 4$ and produced exactly one additional surface (the Kulkarni surface) for $g= 3\mod 4$. In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

References

Robert D. M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. Amer. Math. Soc. 131 (1968), 398–408. MR 222281, DOI 10.1090/S0002-9947-1968-0222281-6
S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa, and G. Gromadzki, Symmetries of Riemann surfaces on which $\textrm {PSL}(2,q)$ acts as a Hurwitz automorphism group, J. Pure Appl. Algebra 106 (1996), no. 2, 113–126. MR 1372846, DOI 10.1016/0022-4049(94)00065-4
E. Bujalance and A. F. Costa, A combinatorial approach to the symmetries of $M$ and $M-1$ Riemann surfaces, Discrete groups and geometry (Birmingham, 1991) London Math. Soc. Lecture Note Ser., vol. 173, Cambridge Univ. Press, Cambridge, 1992, pp. 16–25. MR 1196912, DOI 10.1017/CBO9780511565793.004
Gromadzki G.: Groups of Automorphisms of Compact Riemann and Klein Surfaces. Habilitazionschrift. University Press WSP Bydgoszcz (1993).
Harnack A.: Über die Vieltheiligkeit der ebenen algebraischen Kurven. Math. Ann. 10 (1876), 189–199.
A. H. M. Hoare and D. Singerman, The orientability of subgroups of plane groups, Groups—St. Andrews 1981 (St. Andrews, 1981) London Math. Soc. Lecture Note Ser., vol. 71, Cambridge Univ. Press, Cambridge, 1982, pp. 221–227. MR 679163, DOI 10.1017/CBO9780511661884.014
Hurwitz A.: Über algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41 (1893), 402–442.
Ravi S. Kulkarni, A note on Wiman and Accola-Maclachlan surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 1, 83–94. MR 1127698, DOI 10.5186/aasfm.1991.1615
A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961), 90–96 (1961). MR 146724, DOI 10.1017/S2040618500034365
Macbeath A. M.: Discontinuous groups and birational transformations. Proc. of Dundee Summer School, Univ. of St. Andrews (1961).
A. M. Macbeath, Action of automorphisms of a compact Riemann surface on the first homology group, Bull. London Math. Soc. 5 (1973), 103–108. MR 320301, DOI 10.1112/blms/5.1.103
C. Maclachlan, A bound for the number of automorphisms of a compact Riemann surface, J. London Math. Soc. 44 (1969), 265–272. MR 236378, DOI 10.1112/jlms/s1-44.1.265
Kenji Nakagawa, On the orders of automorphisms of a closed Riemann surface, Pacific J. Math. 115 (1984), no. 2, 435–443. MR 765199, DOI 10.2140/pjm.1984.115.435
S. M. Natanzon, Automorphisms of the Riemann surface of an $M$-curve, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 82–83 (Russian). MR 509395
D. Singerman, Automorphisms of compact non-orientable Riemann surfaces, Glasgow Math. J. 12 (1971), 50–59. MR 296286, DOI 10.1017/S0017089500001142
Weichold G.: Über symmetrische Riemanns’che Flächen und die Periodicitäsmoduln der zugehörin Abel’schen Normalintegrale erster Gattung. Zeitschrift für Math. und Phys. 28 (1883), 321–351.