Cusps of Hilbert modular varieties

TL;DR

This paper characterizes which manifolds can appear as cusp cross-sections of Hilbert modular varieties, providing a complete criterion and revealing that all Sol 3-manifolds can occur in the surface case, with some limitations in higher dimensions.

Contribution

It offers a necessary and sufficient condition for a manifold to be a cusp cross-section of a Hilbert modular variety, especially classifying Sol 3-manifolds in this context.

Findings

All Sol 3-manifolds are cusp cross-sections of Hilbert modular surfaces.

Not all Sol 3-manifolds can be cusp cross-sections of 1-cusped Hilbert modular surfaces.

Obstructions to geometric bounding are identified.

Abstract

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3-manifold is diffeomorphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3-manifolds that cannot arise as a cusp cross-section of a 1-cusped nonsingular Hilbert modular surface.