Sofic groups fulfill determinant conjecture. 

This implies in particular that there exists a natural constant $c$ such that given a matrix $M$ over the integral  group ring of a given sofic group $G$, we have that
$$
|\exp(-cM) - \dim\ker_{vN} M| < \frac{1}{3}.
$$


This can be used to show that some problems about the group are decidable. Suppose a group $G$ is torsion-free, has decidable word problem, fulfills Atiyah conjecture, and is sofic. Then there is an algortihm which decides whether a given matrix $M$ over the integral group ring has non-trivial kernel, as an operator on $[l^2(G)]^{\dim M}$. 

Indeed, given $M$ it's easy to bound it's $l^2$ norm and based on this to decide how many terms in $\exp(-cM)$ have to be computed in order to be less than $\frac{1}{6}$ from the actual value of $\exp(-cM)$. Call this approximation $A$. Now, because $G$ is torsion free and fulfills Atiyah conjecture we know that $\dim\ker_{vN} M$ is an integer, and it's equal to $0$ iff $M$ has trivial kernel. So $M$ kas trivial kernel if and only if $A<\frac{1}{2}$

Similar algorithm works if a group has bounded torsion, since $\frac{1}{3}$ in the first equation can be exchanged with any postivie real number. I seem to have read that there exist Tarski monsters with decidable word problem. That means that in principle :-) one could try to show that there's no such algorithm for these Tarski monsters and arrive at the conclusion that either these monsters are non-sofic or they don't fulfill Atiyah conjecture.