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### Which conjectures are proved for sofic groups? [closed]

Which conjectures about groups are resolved in case of sofic groups?
I know two examples:
Kaplansky's direct finiteness conjecture (proved by Gabor Elek).
Some versions of Ornstein's isomorphism ...

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### Group ring and left zero divisor II

Let $K$ be a finite field and $G$ be a discrete group.
Is it true that for every $a=e+a_1+\ldots+a_n,b=e+b_1+\ldots+b_m\in K[G]$ with $b_i\neq e,a_j\neq e$ the condition $ab=0$ implies $ba=0$?
...

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### Zipper action of a discrete group.

A discrete group $\Gamma$ has zipper action if there is a set $X$ and an action of $\Gamma$ on $X$ (say left-action) and a subset $Z\subseteq X$ such that
for every $g \in \Gamma$: $|gZ\Delta Z|<...

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### Properties of a non-sofic group

This question is, essentially, a comment of Mark Sapir. I think it deserves to be a question.
A countable, discrete group $\Gamma$ is $sofic$ if for every $\epsilon>0$ and finite subset $F$ of $\...