A hyperbolic counterpart to Rokhlin's cobordism theorem

Abstract

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 $n\ge 2$, 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 $n\ge 2$.

16 pages, 4 figures, 3 tables; to appear in Int. Math. Res. Notices; ancillary file available at https://doi.org/10.7910/DVN/N0DP7R