2017/01/10 by Donald I. Cartwright, Cartwright, Donald I., Tim Steger +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Mathematics and Applications #math.GR #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1701.02452
arxiv created 2017/01/10 · openalex publication_date 2017/01/10 · arxiv updated 2017/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In order to enumerate the fake projective planes, as announced in~\citeCS, we found explicit generators and a presentation for each maximal arithmetic subgroup Γ of~PU(2,1) for which the (appropriately normalized) covolume equals~1/N for some integer~N≥1. Prasad and Yeung \citePY1,PY2 had given a list of all such Γ (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(\C2) in~\C2. Our main results here give criteria which ensure that the computer search has found sufficiently many elements of~Γ to generate Γ, and describes a family of relations amongst the generating set sufficient to give a presentation of~Γ. We give an example illustrating details of how this was done in the case of a particular~Γ (for which N=864). While there are no fake projective planes in this case, we exhibit a torsion-free subgroup~Π of index~N in~Γ, and give some properties of the surface~Π\backslash B(\C2).