Census of the Complex Hyperbolic Sporadic Triangle Groups

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic triangle groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and for each group in the list we give a conjectural group presentation, as well as a list of cusps and generators for their stabilisers. We describe strong evidence for these conjectural statements, showing that their validity depends on the solution of reasonably small systems of quadratic inequalities in four variables.