Cusp cross-section phenomena for arithmetic hyperbolic manifolds

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 $n\ge 32$. 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 $n\ge 24$ 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.