Abstract
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).
Citation
M. A. Bridson. D. B. McReynolds. A. W. Reid. R. Spitler. "Absolute profinite rigidity and hyperbolic geometry." Ann. of Math. (2) 192 (3) 679 - 719, November 2020. https://doi.org/10.4007/annals.2020.192.3.1
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