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Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). The Cayley complex of the finite presentation is a $2$-complex where the $1$-skeleton is the Cayley graph, and the $2$-cells are given by the relations (here is a nice YouTube video on it by Daniel Tubbenhauer).

Let us call Cayley $n$-complex of a finite presentation, the corresponding Cayley graph if $n=1$, and the corresponding Cayley complex if $n=2$, as mentioned above.

Question: What about a Cayley $n$-complex made from a finite presentation for $n>2$ (whose $r$-skeleton is a Cayley $r$-complex, for $r<n$)?

I am looking for obstructions preventing such a generalization to all the torsion-free finitely presented groups, and/or references discussing such a generalization to all or a large class of them.

Here is my effort to see what such a generalization might look like.

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    $\begingroup$ Isn't what you are looking for just the $n$-skeleton of a model for $EG$? $\endgroup$
    – IJL
    May 7, 2022 at 15:16
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    $\begingroup$ I don't believe any general approach is known for constructing a model of EG from a finite presentation and Stallings for example early on have a finitely presented group with no model of BG having a finite 3-skeleton. $\endgroup$
    – Benjamin Steinberg
    May 7, 2022 at 19:49
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    $\begingroup$ Stallings paper is jstor.org/stable/2373106. Bieri later gave similar examples where the n-skeleton of BG is finite but not the n+1 skeleton and Bestvina-Brady groups show things are very messy. So it seems hard to believe a finite presentation can lead to anything ask that nice. $\endgroup$
    – Benjamin Steinberg
    May 7, 2022 at 20:18
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    $\begingroup$ About the linked post: I think this "morally" makes sense, but is weak contractibility really that easy? In the Cayley 2-complex it's not true that every embedding of $S^1$ is homotopic to the border of the realization of an irreducible 2-block, that would be like saying every relator in the group is a conjugate of a defining relator (when really you have to use all products of conjugates of defining relators). Similar issues come up for higher $\pi_n$ too. I'm not saying I doubt that this complex is weakly contractible, I'm just saying, I think it's pretty hard. $\endgroup$
    – Matt Zaremsky
    May 8, 2022 at 11:58
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    $\begingroup$ @SebastienPalcoux: My point was that it is very hard to say what one means by good. I just had a quick look at your idea of irreducible $n$-blocks. In the case $n=1$ it has something in common with Brian Bowditch's idea of a taut loop in the Cayley graph (see his article `Continuously many quasi-isometry classes of 2-generator groups'). For larger $n$ I think there is a problem: why should (for example) a 2-block be a 2-sphere rather than say a 2-torus or some other surface? $\endgroup$
    – IJL
    May 9, 2022 at 8:18

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