Finding generators and relations for groups acting on the hyperbolic ball

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~$N\ge 1$. 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~$Π\backslash B(\C^2)$.