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

Question: What about a generalization of the Cayley complex to higher dimensions, but still given by the finite presentation (in particular, the $1$-skeleton is its Cayley graph, and the $2$-skeleton is its Cayley complex)?

I am looking for obstructions preventing such a generalization to all the 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.