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A very basic question that seems to be unsolved: is the class of sofic/hyperlinear groups closed under semi-direct product?

In case of sofic groups the following restricted version is well-known: if $N$ is sofic and $H$ is amenable, then any semidirect product $N\rtimes H$ turns out to be sofic. Is the analogue true for hyperlinear $N$'s?

Thanks in advance for any comment,


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Ops my pc is little crazy today and symbols do not appear. Hope you can see them. – Valerio Capraro May 11 '11 at 9:19
up vote 5 down vote accepted

I think that this is known in the operator algebra commmunity, but it is also a consequence of the proof in the sofic case, which was obtained in

Elek, Gábor, Szabó, Endre, On sofic groups. J. Group Theory 9 (2006), no. 2, 161–171.

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Many thanks. You are anyway talking about the "restricted" version.. – Valerio Capraro May 11 '11 at 14:25
Right. General crossed products are too difficult. – Andreas Thom May 11 '11 at 14:54
The point is that they use the definition with the $(F,\varepsilon)-almost-morphisms$ and this definition makes trivial the higher cardinality case. I like much more the following definition: a group is hyperlinear if it admits a monomorphism $\theta$ into $U(R^\omega)$ such that $\tau(\theta(g))=0$ for all $g\neq1$. In this case it does not seem to me that Elek and Szabo's proof is applicable.. It might be trivial too, please forgive me if it is. – Valerio Capraro May 15 '11 at 19:54

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