Nielsen extensions of Riemann surfaces

Abstract

Let B be a Riemann surface of finite type (p , % , ffi), that is, a sphere with p handles, zr, punctures and zz holes. We assume that m ) 0 and 6p-612n*3m>O (so that B is not a simply or doubly connected plane domain). Nielsen assigned to B a subdomain Bo c B which is a deformation retract of S and again a Riemann surface of t54re (p , n , m); we call So the Nielsen kernel of B . It turns out that every B is the Nielsen kernel of a Riemann surface B, , called the Nielsen extension of B . The purpose of this note is to record. some simple but useful properties of the Nielsen extension. We represent B as UIG where [/ denotes the upper half-plane and G a torsion-free n'uchsian group of the second kind. (The fact that G is determined only up to a conjugation in the real Mbius group will cause no difficulties.) There is an infinite set B of open intervals ,[ on the extended