A Hyperbolic Counterpart to Rokhlin's Cobordism Theorem

The purpose of the present paper is to prove existence of super-exponentially many compact orientable hyperbolic arithmetic n-manifolds that are geometric boundaries of compact orientable hyperbolic (n+1)-manifolds, for any n 2, thereby establishing that these classes of manifolds have the same growth rate with respect to volume as all compact orientable hyperbolic arithmetic n-manifolds. An analogous result holds for non-compact orientable hyperbolic arithmetic n-manifolds of finite volume that are geometric boundaries for n 2.