Automorphisms groups of genus three Riemann surfaces

Abstract

In this work we are going to study the automorphisms of compact non-hyperelliptic Riemann surfaces. In particular, we are going deeply analyse the surfaces of genus three. For such surfaces of genus greater than one, the automorphisms group is finite, and, as a matter of fact, we have a formula who establishes an upper limit on the cardinality of the group depending on the genus of the surface. This formula was found by Hurwitz, and it tells us that the number of automorphisms of a Riemann surface of genus g is finite and bounded by 84( $g − 1$). This upper bound can not be improved in general, as it is reached for some cases.