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Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact,torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinitecompletions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan andgives the first examples nonisomorphic lattices in a semisimple Lie group ofreal rank one with isomorphic profinite completions, answering two questions ofA. Reid.

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