The combinatorial Riemann mapping theorem (Q1343832)

The combinatorial Riemann mapping theorem supplies a surface with local quasi-conformal coordinates compatible with given combinatorial data. Suppose that there is given a conformal sequence of locally finite covers of a metric surface by compact connected sets. Then there is a constant \(k\), depending on the sequence, satisfying the following condition. If \({\mathfrak R}\) is any ring in the surface, then there is a metric on \({\mathfrak R}\) which makes \({\mathfrak R}\) isometrically, a right circular cylinder and in which classical moduli and asymptotic approximate moduli are \(k\)- comparable. Thus, the content of the theorem is the existence of local analytic coordinates for which classical modulus is approximated by asymptotic combinatorial modulus. The theorem could be used to show that certain negatively curved groups have constant curvature.