Estimates for the number of automorphisms of a Riemann surface (Q1375033)

The Hurwitz formula, \[ N \leq 84(g-1), \] estimates the maximal number \(N\) of automorphisms of a Riemann surface of genus \(g>1\). This estimate is sharp, and the equality is realized only for arithmetic Riemann surfaces (i.e. Riemann surfaces uniformized by arithmetic Fuchsian groups). The author improves the estimate for the case of non-arithmetic Riemann surfaces: \[ N \leq 157(g-1)/7. \] He shows that this estimate is sharp. It is attained for infinitely many mutually non-homeomorphic Riemann surfaces. Among them, the surface of genus 50 is the one of minimum genus.