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 $$|\exp(-cM) tr_{vN} \exp(-cM) - \dim\ker_{vN} dim_{vN}\ker M| < \frac{1}{3}.$$
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 its $l^2$ norm and based on this to decide how many terms in $\exp(-cM)$ tr_{vN}\exp(-cM)$have to be computed in order to be less than$\frac{1}{6}$from the actual value of$\exp(-cM)$. tr_{vN} \exp(-cM)$. Call this approximation $A$. a$(it can be computed since the word problem is decidable). Now, because$G$is torsion free and fulfills Atiyah conjecture, we know that$\dim\ker_{vN} \dim_{vN}\ker M$is an integer, and it's equal to$0$iff$M$has trivial kernel. So$M$kas trivial kernel if and only if$Aa<\frac{1}{2}$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. 1 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.