Examples of hyperlinear groups without factorization property

TL;DR

This paper presents examples of hyperlinear groups that lack the factorization property, including a sofic Kazhdan group not residually finite, and discusses related properties and counterexamples in group theory.

Contribution

It provides the first known examples of hyperlinear groups without the factorization property and clarifies misconceptions about subgroup properties within Lie groups.

Findings

A hyperlinear group without the factorization property is constructed.

A sofic Kazhdan group not residually finite is demonstrated.

An example of a group not initially subamenable but hyperlinear is given.

Abstract

In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgroup of a connected finite-dimensional Lie group.