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Symmetries of Accola-Maclachlan
and Kulkarni surfaces


Authors: S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa and G. Gromadzki
Journal: Proc. Amer. Math. Soc. 127 (1999), 637-646
MSC (1991): Primary 14H45, 14E09, 14H30
DOI: https://doi.org/10.1090/S0002-9939-99-04534-7
MathSciNet review: 1468184
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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.


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Additional Information

S. A. Broughton
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Email: allen.broughton@rose-hulman.edu

E. Bujalance
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Email: eb@mat.uned.es

A. F. Costa
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Email: acosta@mat.uned.es

J. M. Gamboa
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Email: jmgamboa@eucmax.sim.ucm.es

G. Gromadzki
Affiliation: Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana 47803
Email: greggrom@mat.uned.es

DOI: https://doi.org/10.1090/S0002-9939-99-04534-7
Received by editor(s): November 15, 1995
Received by editor(s) in revised form: June 5, 1997
Additional Notes: The second and third authors were partially supported by DGICYT PB 95-0017 and CEE-CHRX-CT93-0408.
The fourth author was partially supported by DGICYT PB 95-0354 and CEE-CHRX-CT93-0408
The fifth author was partially supported by the Pedagogical University of Bydgoszcz.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society

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