Hybrid lattices and thin subgroups of Picard modular groups

TL;DR

This paper introduces a hybridization method to construct subgroups of PU(n,1) from pairs of lattices, revealing conditions under which these hybrids are lattices or thin subgroups, with specific results for Picard modular groups.

Contribution

It demonstrates that certain hybrid groups derived from Picard modular groups are lattices or thin subgroups, depending on the parameter d, providing new insights into subgroup structures.

Findings

Hybrid of pairs of Fuchsian subgroups is a lattice for d=1 and d=7.

Hybrid is a geometrically infinite thin subgroup for d=3.

Identifies conditions for hybrid groups to be lattices or thin subgroups.

Abstract

We consider a certain hybridization construction which produces a subgroup of ${\rm PU}(n,1)$ from a pair of lattices in ${\rm PU}(n-1,1)$. Among the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$, we show that the hybrid of pairs of Fuchsian subgroups ${\rm PU}(1,1,\mathcal{O}_d)$ 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.