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Lattices in $ \operatorname{PU}(n,1)$ that are not profinitely rigid


Author: Matthew Stover
Journal: Proc. Amer. Math. Soc. 147 (2019), 5055-5062
MSC (2010): Primary 11F06, 20E18, 20H10; Secondary 11G18, 14G35, 20F67
DOI: https://doi.org/10.1090/proc/14763
Published electronically: September 20, 2019
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Abstract: Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $ \operatorname {PU}(n,1)$ with isomorphic profinite completions for all $ n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples of nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.


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Additional Information

Matthew Stover
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: mstover@temple.edu

DOI: https://doi.org/10.1090/proc/14763
Received by editor(s): August 30, 2018
Published electronically: September 20, 2019
Additional Notes: This material is based upon work supported by Grant Number 523197 from the Simons Foundation/SFARI.
Communicated by: Ken Bromberg
Article copyright: © Copyright 2019 American Mathematical Society