We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ with $\omega^2+\omega+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

中文翻译：

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绝对极限刚性和双曲几何

我们构造了在绝对意义上非常严格的算术克莱因群：每个群都通过其有限商集与所有其他有限生成的剩余有限群区分开来。Bianchi 群 $\mathrm{PSL}(2,\mathbb{Z}[\omega])$ 与 $\omega^2+\omega+1=0$ 在这个意义上是刚性的。其他例子包括 $\mathrm{PSL}(2,\mathbb{C})$ 中最小协体积的非均匀格和 Weeks 流形的基本群（最小体积的闭双曲线 $3$-流形） .