Cusp cross-section phenomena for arithmetic hyperbolic manifolds
Duncan McCoy, Connor Sell
TL;DR
The paper addresses which flat $n$-manifolds can occur as cusp cross-sections of arithmetic hyperbolic manifolds and, in particular, which have the unique arithmetic commensurability class (UCC) property. It develops an algebraic framework based on holonomy forms, discriminants $d(f)$, and Hasse– Witt invariants $\varepsilon_p(f)$ to classify cusp cross-sections via projective equivalence and then builds explicit flat manifolds across dimensions using toral extensions, products, and group actions. The main contributions are: (i) existence of orientable flat $n$-manifolds with the UCC property for all $n\ge 32$, (ii) unbounded growth in the number of commensurability classes containing cusp cross-sections with the UCC property, (iii) orientable non-arithmetic pairs existing in dimensions $n\ge 24$, with concrete constructions in dimensions $n\equiv 0,2,3\pmod 4$. These results map the landscape of cusp cross-sections in arithmetic hyperbolic geometry and provide explicit families for further arithmetic and geometric study, highlighting how holonomy data governs arithmetic realizability of cusp cross-sections.
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.