Let $G$ be a finitely presented group. The [Cayley graph][1] 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][2] 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][3] is a nice YouTube video on it by [Daniel Tubbenhauer][4]).

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][5] is my effort to see what such a generalization might look like.


  [1]: https://en.wikipedia.org/wiki/Cayley_graph
  [2]: https://en.wikipedia.org/wiki/Cayley_complex
  [3]: https://youtu.be/uGc4tGDtpRc
  [4]: https://www.dtubbenhauer.com/
  [5]: https://math.stackexchange.com/q/478588/84284