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Suppose I have an expanding family of Schreier graphs $Z_n=\text{Sch}(G_n,S_n,X_n)$ of groups $G_n=\underbrace{G\wr\ldots\wr G}_{\text{$n$ times}}$ acting on sets $S_n=S^n$ by generating sets $X_n$, such that the generating sets are consistent, in the sense that the restriction of $X_n$ to $G_m$ for $m<n$ is exactly $X_m$. For a more concrete picture, think of the wreath product $G_n$ as acting on a $|S|$-regular tree by permuting the children of each vertex according to permutations in $G$. (If you're familiar with this language, notice that this means that we have a covering map $Z_n\to Z_m$ whenever $m<n$).

It seems that there should be some nice infinite object which is the limit of these graphs, and has some property related to the relative version of property $(\tau)$. More specifically, if we take $G_\infty$ to be the group of symmetries of the infinite $|S|$-regular tree where each automorphism is allowed to permute the children of a vertex using only permutations in $G$, if $\varphi_n:G_\infty\to G_n$ are the restriction maps to finite levels of the tree, and $X_\infty$ are the elements of $G_\infty$ that project to $X_n$ under $\varphi_n$, the graphs $Z_n$ are precisely Schreier graphs of $\left<X_\infty\right>/\ker\varphi_n$. If they were Cayley graphs, this would mean that $\left<X_\infty\right>$ has property $(\tau)$ with respect to the normal subgroups $\{\ker\varphi_n\}$. (also, see section 8 of this paper for context). I wouldn't be surprised if someone has already worked out what this object should be - any ideas for references to look at?

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    $\begingroup$ What they get in the linked paper is some property $\tau$ relative to a given family of finite index subgroup, not the "absolute" property $\tau$ (with respect to all finite index subgroups). $\endgroup$
    – YCor
    Oct 24, 2016 at 4:48
  • $\begingroup$ @YCor: also thanks for giving this quesiton better tags $\endgroup$
    – amakelov
    Oct 25, 2016 at 0:17

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