Geodesic
Absolute profinite rigidity and hyperbolic geometry
M. R. Bridson1 ; D. B. McReynolds2 ; A. W. Reid3 ; R. Spitler4
1 Mathematical Institute, University of Oxford, Oxford, UK
2 Department of Mathematics, Purdue University, West Lafayette, IN, USA
3 Department of Mathematics, Rice University, Houston, TX, USA
4 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada
Annals of mathematics, Tome 192 (2020) no. 3, pp. 679-719

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

DOI : 10.4007/annals.2020.192.3.1

M. R. Bridson 1 ; D. B. McReynolds 2 ; A. W. Reid 3 ; R. Spitler 4

1 Mathematical Institute, University of Oxford, Oxford, UK
2 Department of Mathematics, Purdue University, West Lafayette, IN, USA
3 Department of Mathematics, Rice University, Houston, TX, USA
4 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada 
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M. R. Bridson; D. B. McReynolds; A. W. Reid; R. Spitler. Absolute profinite rigidity and hyperbolic geometry. Annals of mathematics, Tome 192 (2020) no. 3, pp. 679-719. doi: 10.4007/annals.2020.192.3.1

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