It is well-known that the cusp cross-sections of finite-volume, cusped hyperbolic n-manifolds are flat, compact (n-1)-manifolds. In 2002, Long and Reid proved that each of the finitely many homeomorphism classes of flat, compact (n-1)-manifolds occur as the cusp cross-section of some arithmetic hyperbolic n-orbifold; the orbifold was upgraded to a manifold by McReynolds in 2004.

There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This thesis provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains a representative with a given cusp type. In particular, for three of the six cusp types, we provide infinitely many examples of commensurability classes that contain no manifolds with cusps of the given type; no such examples were previously known for any cusp type in any dimension. Further, we extend this result to find commensurability classes of hyperbolic 5-manifolds that avoid some compact, flat 4-manifolds as cusp cross-sections, and classes of non-arithmetic manifolds in both dimensions that avoid some cusp types.