A discrete group $\Gamma$ has **zipper** action if there is a set $X$ and an action of $\Gamma$ on $X$ (say left-action) and a subset $Z\subseteq X$ such that

- for every $g \in \Gamma$: $|gZ\Delta Z|< \infty$
- for every $r>0$ the set $\{g\in \Gamma : |gZ\Delta Z|\leq r\}$ is finite.

Is it true that if $\Gamma$ has zipper action then $\Gamma$ is sofic?

Definition of sofic group can be found for instance in the survey: http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.3968v8.pdf