What are the "simplest" examples of countable groups that are not known to be sofic?
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$\begingroup$ I think Thompson's groups are not known to be sofic. See Pestov's survey: google.com/… $\endgroup$– Ian AgolCommented Feb 16, 2014 at 5:56
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The simplest candidate I know of is Higman's group
$\langle a,b,c,d\mid a^b=a^2, b^c=b^2, c^d=c^2, d^a=d^2\rangle$
(where, as usual, $a^b$ means $b^{-1}ab$). Terry Tao wrote a nice blog post about it here.
Residually finite and amenable groups are both known to be sofic. As Tao explains, Higman's group has no finite quotients (hence is 'highly non-residually finite') and a non-abelian free subgroup (hence is certainly non-amenable).
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$\begingroup$ Thank you very much! Are there any non-residually finite non-amenable groups that are known to be sofic? $\endgroup$– VladimirCommented Feb 10, 2014 at 18:03
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4$\begingroup$ Soficity is preserved under direct products, so just take a direct product of non-amenable and non residually finite sofic groups, such as a free group and a Baumslag-Solitar group like $\langle x,y \mid y^{-1}x^2y=x^3 \rangle$. $\endgroup$ Commented Feb 10, 2014 at 18:21
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1$\begingroup$ A more recent reference appears here: arxiv.org/abs/1512.02135 $\endgroup$– Ian AgolCommented Aug 19, 2016 at 4:51