On extremal Riemann surfaces and their uniformizing Fuchsian groups

Compact hyperbolic surfaces of given genus g containing discs of the maximum radius have been studied from various points of view. In this paper we connect these different approaches and observe some properties of the Fuchsian groups uniformizing both compact and punctured extremal surfaces. We also show that extremal surfaces of genera g = 2, 3 may contain one or several extremal discs, while an extremal disc is necessarily unique for g greater than or equal to 4. Along the way we also construct explicit families of extremal surfaces, one of which turns out to be free of automorphisms.