Examples of hyperlinear groups without factorization property

Understand

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.