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Which conjectures about groups are resolved in case of sofic groups?

I know two examples:

  1. Kaplansky's direct finiteness conjecture (proved by Gabor Elek).
  2. Some versions of Ornstein's isomorphism theorem (proved by Lewis Bowen).

What other examples are there?

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    $\begingroup$ Which conjectures about groups do you know? $\endgroup$
    – Yemon Choi
    Mar 12, 2016 at 0:40
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    $\begingroup$ I would add to the list Gottschalk's conjecture (in a very general version) by Gromov (who defined sofic groups) $\endgroup$
    – Simone Virili
    Mar 12, 2016 at 2:26

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