Seminário de Geometria Algébrica
Título: Classifying compact Riemann surfaces by number of symmetries
Resumo: Let X be a compact Riemann surface of genus g at least 2. Then an automorphism group G of X is finite and its order is upper bounded by the celebrated Hurwitz bound
|G| <= 84(g-1).
If one imposes additional conditions on the structure of G, then tighter bounds may hold; for instance, if G is cyclic, then
|G| <= 4g+2.
Curves that are extremal with respect to the latter bound exist for every genus and were classified by Nakagawa. Later, Kulkarni took the classification problem further and studied the classification of Riemann Surfaces having 4g+2 automorphisms, with no restriction on the structure of such a group.
This talk addresses the problem of classifying Riemann surfaces that admit 3g and 3g+3 automorphisms. For infinite families of genera, we classify them, as well as provide results on their Jacobians and full automorphism groups.