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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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Official URL: http://www.jstor.org/stable/2046834

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.