A hyperbolic lattice in each dimension with Zariski-dense surface subgroups

TL;DR

This paper constructs specific nonuniform arithmetic lattices in hyperbolic spaces of dimensions three and higher that contain surface subgroups which are dense in the Zariski topology, advancing understanding of lattice subgroup structures.

Contribution

It demonstrates the existence of Zariski-dense surface subgroups within nonuniform arithmetic lattices in all dimensions n ≥ 3.

Findings

Existence of Zariski-dense surface subgroups in lattices

Construction of hyperbolic lattices with dense surface subgroups

Extension of known lattice subgroup properties to higher dimensions

Abstract

For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.