Examples of hyperlinear groups without factorization property

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 from [ES05]. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in [Kir94], Corollary 1.2 (v) $\Rightarrow$ (i), and [Kir93], Corollary 7.3 (iii), 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.