Profinite Non-Rigidity of Arithmetic Lattices and the K\"ahler Property

Abstract

We study the profinite non-rigidity of arithmetic lattices and its implications for the K\"ahler property. In the first part, we characterize the absolute Dynkin types that admit non-isomorphic real forms of higher-rank Lie groups containing torsion-free arithmetic lattices with isomorphic profinite completions. In the second part, as a geometric application, we answer a question asked independently by Arapura and Libgober, by showing that the K\"ahlerness of finitely presented, residually finite groups is not determined by its profinite completion.