Cusps of Hyperbolic 4-manifolds

The cross-sections of the cusps of (real) hyperbolic 4-manifolds are diffeomorphic to compact, flat 3-manifolds.  Long and Reid have proven that all six orientable flat 3-manifolds occur as cusp cross-sections of hyperbolic manifolds, but further questions can be asked about the conditions under which each cross-section can occur.  I will discuss a result that uses arithmetic methods to show that three of the flat manifolds occur in every commensurability class of arithmetic hyperbolic 4-manifolds, and three do not.