Infinite circle packings on surfaces with conical singularities

Abstract

We show that given an infinite triangulation K of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than π at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of K. Moreover, if K is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation K and a conformal class X such that there are infinitely many conical hyperbolic structures in the conformal class X with a circle packing in the combinatorics of K. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.