On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups (Q6613275)

A lattice \(\Gamma \subset \mathrm{PU}(2,1)\) is torsion-free if and only if the quotient map \(\mathbb{H}^{2}_{\mathbb{C}} \rightarrow \mathbb{H}^{2}_{\mathbb{C}}/\Gamma\) is an unramified covering (equivalently, the quotient \(\mathbb{H}^{2}_{\mathbb{C}}/ \Gamma\) is a manifold with charts given by local inverses of the quotient map from \(\mathbb{H}^{2}_{\mathbb{C}}\)). A lattice is neat if it is torsion-free and every parabolic element in the group can be realized by a unipotent matrix.\N\NIn the paper under review, the author studies several explicit finite index subgroups in the complex hyperbolic lattice triangle groups and shows that some of them are neat, some of them have positive first Betti number and some of them have a homomorphisms onto a non-abelian free group. For some lattice triangle groups, he determines the minimal index of a neat subgroup (see [\textit{A. Borel}, Introduction aux groupes arithmétiques. Paris: Hermann \& Cie (1969; Zbl 0186.33202)]). Finally, he answers a question raised by \textit{M. Stover} [Geom. Dedicata 157, 239--257 (2012; Zbl 1301.22006)] and describes an infinite tower of neat ball quotients all with a single cusp.