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Large automorphism groups of hyperelliptic Klein surfaces

Bujalance, E. and Etayo Gordejuela, J. Javier (1988) Large automorphism groups of hyperelliptic Klein surfaces. Proceedings of the American Mathematical Society, 103 (3). pp. 679-686. ISSN 0002-9939

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Abstract

A bordered Klein surface of algebraic genus p has at most 12(p-1) automorphisms and this is attained for infinitely many values of p. Furthermore, for an infinity of values of p, the largest group of automorphisms of such a surface is $4(p+1)$ or 4p depending on whether the surface is orientable or not [{\it C. L. May}, Pac. J. Math. 59, 199- 210 (1975) and Proc. Am. Math. Soc. 63, 273-280 (1977]. \par Here the authors examine such surfaces which are additionally hyperelliptic and have automorphism groups of order exceeding 4(p-1). Using their characterization of hyperelliptic Klein surface via non- Euclidean crystallographic groups [Q. J. Math., Oxf. II. Ser. 36, 141-157 (1985)] the authors determine these automorphism groups, which are all dihedral or direct sums of a dihedral group and a cyclic group of order 2, and the corresponding topological type of the surface.

Item Type:Article
Uncontrolled Keywords:Fuchsian groups and their generalizations; Curves; Compact Riemann surfaces and uniformization
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:15766
References:

N. I. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Math., vol. 219, Springer-Verlag, Berlin-Heidelberg-New York, 1971.

E. Bujalance, Proper periods of normal NEC subgroups with even index, Rev. Mat. Hisp.-Amer. (4) 41 (1981), 121-127.

E. Bujalance and J. J. Etayo, Hyperelliptic Klein surfaces with maximal symmetry, Proc. War- wick and Durham Symposia 1984, London Math. Soc. Lecture Note Series, 112, 1986, pp. 289-296.

E. Bujalance, J. J. Etayo, and J. M. Gamboa, Hyperelliptic Klein surfaces, Quart. J. Math. Oxford Ser. (2) 361985, pp. 141-157.

— Group of automorphisms of hyperelliptic Klein surfaces of genus three, Michigan Math. J.33 (1986), 55-74.

E. Bujalance and J. M. Gamboa, Automorphisms groups of algebraic curves of Rn of genus 2, Arch. Math.42 (1984), 229-237.

J. A. Bujalance, Normal subgroups of even index of an NEC group, Arch. Math. (to appear).

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C. L. May, Automorphisms of compact Klein surfaces with boundary, Pacific J. Math.59 (1975), 199-210.

--A Bound for the Number of Automorphisms of a Compact Klein Surface with Boundary. Proceedings of the American Mathematical Society Vol. 63, No. 2 (Apr., 1977) pp. 273-280

— Cyclic automorphi3m groups of compact bordered Klein surfaces, Houston J. Math.3 (1977), 395-405.

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