Absolute profinite rigidity and hyperbolic geometry

By Martin R. Bridson, D. McReynolds, Alan W. Reid, R. Spitler. Annals of Mathematics, 2018

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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).

Subfields: Geometry and Topology, Mathematical Physics

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