A hyperbolic counterpart to Rokhlin's cobordism theorem

TL;DR

This paper proves the existence of super-exponentially many hyperbolic arithmetic manifolds that are geometric boundaries of higher-dimensional hyperbolic manifolds, showing they share the same volume growth rate.

Contribution

It establishes a hyperbolic analogue of Rokhlin's cobordism theorem for all dimensions n ≥ 2, demonstrating the abundance of such boundary manifolds.

Findings

Existence of super-exponentially many hyperbolic arithmetic n-manifolds as geometric boundaries.

Same volume growth rate for boundary manifolds and all hyperbolic arithmetic n-manifolds.

Results hold for both compact and finite-volume non-compact hyperbolic manifolds.

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 $n \geq 2$, 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 $n \geq 2$.