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 $n \geq 2$.

中文翻译：

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Rokhlin 的 Cobordism 定理的双曲线对应

本文的目的是证明存在超指数多个紧致可定向双曲算术$n$-流形，它们是紧致可定向双曲$(n+1)$-流形的几何边界，对于任何$n\geq 2$ ，从而确定这些流形类别与所有紧凑可定向双曲线算术 $n$-流形具有相同的体积增长率。对于$n \geq 2$，对于作为几何边界的有限体积的非紧致可定向双曲算术$n$-流形，类似的结果成立。