Cusp types of arithmetic hyperbolic manifolds

Abstract

We establish necessary and sufficient conditions for determining when a flat manifold can occur as a cusp cross-section within a given commensurability class of cusped arithmetic hyperbolic manifolds. This reduces the problem of identifying which commensurability classes of arithmetic hyperbolic manifolds can contain a specific flat manifold as a cusp cross-section to a question involving rational representations of the flat manifold's holonomy group. More generally we show that the holonomy representation provides an obstruction on the quasi-arithmetic manifolds containing a given flat manifold as a cusp cross-section. As applications, we prove that a flat manifold $M$ with a holonomy group of odd order appears as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds if and only if $b_1(M)\geq 3$. We also provide examples of flat manifolds that arise as cusp cross-sections in a unique commensurability class of arithmetic hyperbolic manifolds and exhibit examples of pairs of flat manifolds that can never appear as cusp cross-sections in the same quasi-arithmetic hyperbolic manifold.