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

TL;DR

This paper constructs examples of nonisomorphic, cocompact, torsion-free lattices in complex hyperbolic space with identical profinite completions, challenging previous conjectures and questions in the field.

Contribution

It provides the first known examples of nonisomorphic lattices in rank-one semisimple Lie groups sharing the same profinite completion, using Shimura variety conjugation.

Findings

Disproved Kazhdan's conjecture on profinite rigidity.

First examples of nonisomorphic lattices with isomorphic profinite completions in rank-one groups.

Answered open questions by Reid regarding profinite properties of lattices.

Abstract

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