Absolute profinite rigidity and hyperbolic geometry


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 ${\rm {PSL}}(2,\mathbb {C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).

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M. R. Bridson

Mathematical Institute, University of Oxford, Oxford, UK

D. B. McReynolds

Department of Mathematics, Purdue University, West Lafayette, IN, USA

A. W. Reid

Department of Mathematics, Rice University, Houston, TX, USA

R. Spitler

Department of Mathematics, McMaster University, Hamilton, Ontario, Canada