## Automorphism groups of hyperelliptic Riemann surfaces

### Impacto

Bujalance, E. and Etayo Gordejuela, J. Javier (1987) Automorphism groups of hyperelliptic Riemann surfaces. Kodai Mathematical Journal, 10 (2). pp. 174-181. ISSN 0386-5991

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Official URL: http://projecteuclid.org/euclid.kmj/1138037412

## Abstract

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\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.

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