Automorphism groups of hyperelliptic Riemann surfaces (Q1096018)

If 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 \(G_ 1\) of order 2 and if \(\Gamma_ 1\) is the corresponding subgroup of \(\Gamma\) ', then \(G/G_ 1\) is a group of automorphisms of the sphere \(D/\Gamma_ 1\). 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.