Ideal Triangle Groups, Dented Tori, and Numerical Analysis

Abstract

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is discretely embedded in PU(2, 1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates.