# Zipper action of a discrete group.

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

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This is an interesting question, as it is mentioned in Pestov's survey that it is unknown whether Haagerup groups are all sofic. Groups with zipper actions are all Haagerup, as shown here in a paper of Bruce Hughes: arxiv.org/abs/0804.0610 – Jon Bannon Apr 12 '11 at 18:37
Do you think that there could be relation? What kind of answer do you expect? – Andreas Thom Apr 13 '11 at 12:11
@Andreas: recently I've heard that there is a person who claims that Haagerup p-ty implies soficity, he does not have notes yet, so this can not be checked... But I am not completely sure that one can get such result. if it could be done, then something much simpler (like zipper action) should imply soficity. – Kate Juschenko Apr 13 '11 at 14:33
Can you say who claimed this? This would be really striking. – Andreas Thom Apr 13 '11 at 21:39
For sure it is an open (and interesting!) question. – Alain Valette May 2 '11 at 21:14