Census of the complex hyperbolic sporadic triangle groups

TL;DR

This paper provides a conjectural classification of complex hyperbolic sporadic groups, identifying which are lattices and offering detailed conjectural presentations and stabilizer data, supported by evidence based on quadratic inequalities.

Contribution

It offers the first comprehensive conjectural census of complex hyperbolic sporadic groups, including a list of lattices and their properties, with evidence based on solving quadratic inequalities.

Findings

Finitely many sporadic groups are lattices

Conjectural list of all lattice sporadic groups

Evidence based on quadratic inequalities

Abstract

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic 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 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.