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,


Valerio



Sofic/hyperlinear groups
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. 


