Lattices in 𝑃𝑈(𝑛,1) that are not profinitely rigid

Stover, Matthew

doi:10.1090/proc/14763

Lattices in $\operatorname {PU}(n,1)$ that are not profinitely rigid
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Proc. Amer. Math. Soc. 147 (2019), 5055-5062 Request permission

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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