Finding generators and relations for groups acting on the hyperbolic ball

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~$\Pi\backslash B(\C^2)$.