Properties of a non-sofic group - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:15:15Z http://mathoverflow.net/feeds/question/43824 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43824/properties-of-a-non-sofic-group Properties of a non-sofic group Jon Bannon 2010-10-27T16:16:23Z 2011-05-05T08:01:48Z <p>This question is, essentially, a comment of Mark Sapir. I think it deserves to be a question.</p> <p>A countable, discrete group $\Gamma$ is $sofic$ if for every $\epsilon>0$ and finite subset $F$ of $\Gamma$ there exists an $(\epsilon,F)$-almost action of $\Gamma$. See, for example Theorem 3.5 of the nice survey of Pestov <a href="http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.3968v8.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.3968v8.pdf</a>.</p> <p>Gromov asked whether all countable discrete groups are sofic. It is now widely believed that there should be a counterexample to this.</p> <p>Since most groups are sofic, it would be useful to have a collection of properties that would imply that a group is not sofic...so one can then construct a beast having such properties.</p> <blockquote> <p>What are some abstract properties of $\Gamma$ that would imply $\Gamma$ is not sofic?</p> </blockquote> <p>An open question of Nate Brown asks whether all one-relator groups are sofic. I'd be interested to know what properties of a one-relator group $\Gamma$ would imply that $\Gamma$ is not sofic.</p> http://mathoverflow.net/questions/43824/properties-of-a-non-sofic-group/43829#43829 Answer by Andreas Thom for Properties of a non-sofic group Andreas Thom 2010-10-27T16:28:29Z 2010-10-28T06:04:01Z <p>Let $\Gamma$ be a sofic group. Gabor Elek and Endre Szabo showed <a href="http://arxiv.org/abs/math/0305440" rel="nofollow">here</a>, that for any field $k$ and $a,b \in k[\Gamma]$ with $ab=1$ one has $ba=1$. Hence, coming up with a cleverly chosen group where this fails would provide a counterexample. Note that $k=\mathbb C$ is not a good start since Kaplansky showed long ago that the implication holds for fields of characteristic zero. However, for $k= GF(2)$ one might be lucky.</p> <p>Let us consider $k=GF(2)$ for now. One strategy could be to start with $a = \sum_{g \in F} g$ and $b = \sum_{h \in K} h$ for some finite sets $F,K \subset G$. If $ab=1$, then a number of relations must hold: We quickly convince ourselves that $F$ and $K$ must have an odd number of elements and there exists a self-matching of the set $F \times K \setminus (f,k)$ such that matched pairs $(f',k') \sim (f'',k'')$ satisfy $f'k' = f''k''$ and $f=k^{-1}$ for the special unmatched pair. You can now turn everything around and start with an abstract group with generators $F \cup K$ and relations as above coming from an arbitrarily chosen self-matching. In the group ring of this abstract group, we will have $ab=1$, but why do we have $ba=1$? I was working on this for a while but could not come up with a counterexample. On the other hand, if $F$ and $K$ are large, I cannot believe that $ba=1$ will always hold.</p> http://mathoverflow.net/questions/43824/properties-of-a-non-sofic-group/44286#44286 Answer by Łukasz Grabowski for Properties of a non-sofic group Łukasz Grabowski 2010-10-30T21:30:32Z 2010-10-30T21:37:00Z <p>Sofic groups fulfill determinant conjecture. </p> <p>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 $$|tr_{vN} \exp(-cM) - \dim_{vN}\ker M| &lt; \frac{1}{3}.$$</p> <p>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 $[l^2(G)]^{\dim M}$. </p> <p>Indeed, given $M$ it's easy to bound its $l^2$ norm and based on this to decide how many terms in $tr_{vN}\exp(-cM)$ have to be computed in order to be less than $\frac{1}{6}$ from the actual value of $tr_{vN} \exp(-cM)$. Call this approximation $a$ (it can be computed since the word problem is decidable). Now, because $G$ is torsion free and fulfills Atiyah conjecture, we know that $\dim_{vN}\ker M$ is an integer, and it's equal to $0$ iff $M$ has trivial kernel. So $M$ kas trivial kernel if and only if $a&lt;\frac{1}{2}$</p> <p>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.</p>