Nil 3-manifolds and cusps of complex hyperbolic surfaces

TL;DR

This paper classifies which Nil 3-manifolds can appear as cusps in arithmetic complex hyperbolic 2-manifolds and shows all Nil 3-manifolds can occur in non-arithmetic cases.

Contribution

It provides a classification of Nil 3-manifolds as cusp cross-sections in arithmetic and non-arithmetic complex hyperbolic surfaces, revealing their occurrence patterns.

Findings

Some Nil 3-manifolds occur in all arithmetic classes.

Some Nil 3-manifolds occur only in a single class.

All Nil 3-manifolds occur in non-arithmetic cases.

Abstract

McReynolds showed that every compact Nil 3-manifold occurs as the cusp cross-section of some arithmetic complex hyperbolic 2-manifold. We classify which commensurability classes of cusped, arithmetic, complex hyperbolic 2-manifolds admit cusps with cross-section homeomorphic to a given compact Nil 3-manifold. In particular, there are some Nil 3-manifolds which occur as cusps in every such commensurability class, and some which only occur in a single commensurability class. We also show that every compact Nil 3-manifold occurs as the cusp cross-section of some non-arithmetic complex hyperbolic 2-manifold.