This question is, essentially, a comment of Mark Sapir. I think it deserves to be a question.

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 http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.3968v8.pdf.

Gromov asked whether all countable discrete groups are sofic. It is now widely believed that there should be a counterexample to this.

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.

What are some abstract properties of $\Gamma$ that would imply $\Gamma$ is not sofic?

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.