## Large automorphism groups of hyperelliptic Klein surfaces

### Impacto

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

 PDF Restringido a Repository staff only hasta 31 December 2020. 231kB

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

## 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 Fuchsian groups and their generalizations; Curves; Compact Riemann surfaces and uniformization Sciences > Mathematics > Algebraic geometry 15766 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). A. H. M. Hoare and D. Singerman, The orientability of subgroups of plane groups, London Math. Soc. Lecture Note Series, 71, 1982, pp. 221-227. 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. 26 Jun 2012 10:47 26 Jun 2012 10:47

Repository Staff Only: item control page