Bujalance, E.Etayo Gordejuela, J. Javier2023-06-212023-06-211987COXETER, H. S. M., AND MOSER, W. O. J., "Generators and relations for discrete groups". Ergeb. der Math., 14. Springer. Berlin, etc. 1980 (4th ed.). HARVEY, W. J., "On branch loci in Teichmuller space". Trans. Amer. Math. Soc 153 (1971), 387-399. HURWITZ, A., "Uber algebraische Gebilde mit eindeutigen Transformationen i sich". Math. Ann. 41 (1893), 403-442. KURIBAYASHI, A., "On analytic families of compact Riemann surfaces with non trivial automorphisms". Nagoya Math. J. 28 (1966), 119-165. KURIBAYASHI, I., "Hyperelliptic AM curves of genus three and associated representations". Preprint. MACBEATH, A. M., "Discontinuous groups and birational transformations". Proc Dundee Summer School 1961, 59-75. MACLACHLAN, C., "A bound for the number of automorphisms of a compact Riemann surface".J. London Math. Soc. 44 (1969), 265-272. MACLACHLAN, C., Maximal normal Fuchsian groups. 111. J. Math. 15 (1971), 104 113. MACLACHLAN, C., Smooth coverings of hyperelliptic surfaces. Quart. J. Math Oxford. (2) 22(1971), 117-123. SINGERMAN, D., Symmetries of Riemann surfaces with large automorphism group Math. Ann. 210 (1974), 17-32. SINGERMAN, D., Symmetries and pseudo-symmetries of hyperelliptic ^surf aces Glasgow Math. J. 21 (1980), 39-49. WIMAN, A., Uber die hypereiptischen Curven und diejenigen vom^Geschecht p=3, welche eindeutigen Transformationen in sich zulassen, Bihang Kongl. Svenska Vetenskapsakademiens Handlingar, Stockholm 1895-96.0386-599110.2996/kmj/1138037412https://hdl.handle.net/20.500.14352/64651If G is a group of automorphisms of a hyperelliptic Riemann surface of genus g represented as D/$\Gamma$ where D is the hyperbolic plane and $\Gamma$ a Fuchsian group, then $G\cong \Gamma '/\Gamma$ where $\Gamma$ ' is also a Fuchsian group. Furthermore G contains a central subgroup $G\sb 1$ of order 2 and if $\Gamma\sb 1$ is the corresponding subgroup of $\Gamma$ ', then $G/G\sb 1$ is a group of automorphisms of the sphere $D/\Gamma\sb 1$. Using this and structure theorem for Fuchsian groups the authors determine all surfaces of genus $g>3$ admitting groups G with $o(G)>8(g-1)$. There are two infinite families both corresponding to $\Gamma$ ' being the triangle group (2,4,m) and six other groups.engjournal articlehttp://projecteuclid.org/euclid.kmj/1138037412http://projecteuclid.org/restricted access512.7Classification theory of Riemann surfacesCoveringsfundamental groupSpecial curves and curves of low genusGeometria algebraica1201.01 Geometría Algebraica