Hybrid lattices and thin subgroups of Picard modular groups

Abstract

We consider a certain hybridization construction which produces a subgroup of $\mathrm{PU}(n,1)$ from a pair of lattices in $\mathrm{PU}(n-1,1)$. Among the Picard modular groups $\mathrm{PU}(2,1,{\mathcal{O}}_d)$, we show that the hybrid of pairs of Fuchsian subgroups $\mathrm{PU}(1,1,{\mathcal{O}}_d)$ is a lattice when $d=1$ and $d=7$, and a geometrically infinite thin subgroup when $d=3$, that is an infinite-index subgroup with the same Zariski-closure as the full lattice.

14 pages; 3 figures. Added figures, corrected definition, revised argument in Corollary 2