Asymptotic Equivalence of Automorphism Groups of Surfaces and Riemann-Hurwitz Solutions

An automorphism group of a compact Riemann surface is often described by a tuple $(h; m_1, ..., m_s)$ called its signature which encodes the topological data of the group. There are certain number theoretical conditions on a tuple necessary for it to be the signature of an action: the Riemann-Hurwitz formula is satisfied and each $m_i$ equals the order of a non-trivial group element. Our main result is that asymptotically speaking, satisfaction of these two conditions is sufficient for a tuple to be a signature.