Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension $d\ge 5$ up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest.

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负曲率的同源性和同伦复杂性

为所有维度 $d\ne 3$ 中有限体积的负弯曲流形的扭转同源性的大小提供了线性上限。这扩展了 Gromov 的经典定理。在维度 $3$ 中，与 Betti 数相反，扭转同源性的大小在体积方面是无界的。此外，在 Benjamini--Schramm 拓扑中，存在一系列 $3$ 维双曲流形收敛到 $\mathbb{H}^3$，而其第一个同调中的归一化扭转在 $[0,\infty] 中是密集的$. 在维度 $d\geq 4$ 中，对有限体积的负弯曲流形的数量给出了一些精确的估计，直到同伦，在维度 $d\ge 5$ 中给出了同胚。这些结果基于具有独立意义的有效单纯厚-薄分解。