What are the "simplest" examples of countable groups that are not known to be sofic?



Candidates for non-sofic groups
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](http://terrytao.wordpress.com/2008/10/06/finite-subsets-of-groups-with-no-finite-models/).


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).


