Finding generators and relations for groups acting on the hyperbolic ball

TL;DR

This paper develops methods to find generators and relations for certain arithmetic groups acting on the hyperbolic ball, aiding the enumeration of fake projective planes and analyzing associated complex surfaces.

Contribution

It provides criteria ensuring the sufficiency of computer-generated elements for group presentation and describes a family of relations for these groups, with a detailed example.

Findings

Criteria for computer search completeness

Explicit group presentations obtained

Example of a torsion-free subgroup and surface properties

Abstract

In order to enumerate the fake projective planes, as announced in~\cite{CS}, we found explicit generators and a presentation for each maximal arithmetic subgroup $\bar\Gamma$ of~$PU(2,1)$ for which the (appropriately normalized) covolume equals~$1/N$ for some integer~$N\ge1$. Prasad and Yeung \cite{PY1,PY2} had given a list of all such $\bar\Gamma$ (up to equivalence). The generators were found by a computer search which uses the natural action of $PU(2,1)$ on the unit ball $B(\C^2)$ in~$\C^2$. Our main results here give criteria which ensure that the computer search has found sufficiently many elements of~$\bar\Gamma$ to generate $\bar\Gamma$, and describes a family of relations amongst the generating set sufficient to give a presentation of~$\bar\Gamma$. We give an example illustrating details of how this was done in the case of a particular~$\bar\Gamma$ (for which $N=864$). While there are no fake projective planes in this case, we exhibit a torsion-free subgroup~$\Pi$ of index~$N$ in~$\bar\Gamma$, and give some properties of the surface~$\Pi\backslash B(\C^2)$.