A [Cayley graph][1] is a $1$-dimensional cell complex associated to any presentation of a group $G$, where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (connecting two elements). A [Cayley complex][2] is a $2$-dimensional generalization, where the $2$-cells are given by the relations ([here][3] is a nice YouTube video on it by [Daniel Tubbenhauer][4]).

**Question**: What about a generalization of the Cayley complex to higher dimensions?

I am looking for obstructions preventing such a generalization to all the finitely generated (or 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