Cusps and Commensurability Classes of Hyperbolic 4-Manifolds

TL;DR

This paper establishes criteria for when certain hyperbolic 4-manifolds contain specific cusp types, providing new examples of classes without certain cusp types, advancing understanding of their geometric structures.

Contribution

It offers the first known examples of commensurability classes lacking manifolds with specific cusp types and characterizes when certain cusp types occur in hyperbolic 4-manifolds.

Findings

Identified criteria for cusp types in hyperbolic 4-manifolds

Provided infinitely many classes without certain cusp types

Expanded knowledge of cusp cross-section possibilities

Abstract

There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type.