Cusp cross-section phenomena for arithmetic hyperbolic manifolds

TL;DR

This paper investigates the unique occurrence of certain flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds, constructing examples with the UCC property in all dimensions ≥32 and analyzing their distribution across commensurability classes.

Contribution

It introduces the UCC property for flat manifolds, constructs such manifolds in all dimensions ≥32, and shows the unboundedness of classes with this property, advancing understanding of cusp cross-sections.

Findings

Constructed flat manifolds with the UCC property in all dimensions ≥32.

Proved the number of classes with UCC property is unbounded.

Identified pairs of manifolds that cannot share the same commensurability class.

Abstract

Although every flat manifold occurs as a cusp cross-section in at least one commensurability class of arithmetic hyperbolic manifolds, it turns out that some flat manifolds have the property that they occur as cusp cross-sections in precisely one commensurability class of arithmetic hyperbolic manifolds -- a phenomena which we will refer to as the UCC property. We construct flat manifolds with the UCC property in all dimensions $ n \geq 32 $. We also show that the number of distinct commensurability classes containing cusp cross-sections with the UCC property is unbounded. We also exhibit pairs of manifolds in all dimensions $ n \geq 24 $ that cannot arise as cusp cross-sections in the same commensurability class of arithmetic hyperbolic manifolds. The main tool is previous work of the authors algebraically characterizing when a given flat manifold arises as the cusp cross-section of a manifold in a given commensurability class of arithmetic hyperbolic manifolds.