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August 2022 Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property
M. R. Bridson, A. W. Reid
Michigan Math. J. 72: 25-49 (August 2022). DOI: 10.1307/mmj/20217218


Let Γ be a nonelementary Kleinian group and H<Γ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C).


Dedicated to Gopal Prasad on the occasion of his 75th birthday


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M. R. Bridson. A. W. Reid. "Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property." Michigan Math. J. 72 25 - 49, August 2022.


Received: 28 July 2021; Revised: 20 September 2021; Published: August 2022
First available in Project Euclid: 2 August 2022

Digital Object Identifier: 10.1307/mmj/20217218

Primary: 20H10
Secondary: 20E18

Rights: Copyright © 2022 The University of Michigan


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Vol.72 • August 2022
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