Profinite completions and Kazhdan's property (T)

TL;DR

This paper demonstrates that Kazhdan's property (T) is not detectable solely from a group's profinite completion by constructing two residually finite groups with identical profinite completions but differing in property (T).

Contribution

It provides a counterexample showing property (T) is not a profinite property, answering an open question by Kassabov.

Findings

Constructed two finitely generated residually finite groups with isomorphic profinite completions.

One group has property (T), the other does not.

Property (T) cannot be inferred from profinite completions.

Abstract

We show that property (T) is not profinite, that is, we construct two finitely generated residually finite groups which have isomorphic profinite completions while one admits property (T) and the other does not. This settles a question raised by M. Kassabov.