what is the Cayley complex of dihedral group $D_{4}$? what is the Cayley complex of dihedral group $D_{4}$?
I am aware of Cayley graph of $D_{4}$,can you explain to me how I should I attach 2-cell complexes to the loops to make it covering space?
I can't visualize what is happening,It will be great if you guide me with some hint or help.thank you very much.
 A: Consider the following presentation of the dihedral group of the $4$-gon:
$$
\langle r, s \mid r^4 = s^2 = 1, rs = sr^{-1}\rangle.
$$
Here is the corresponding Cayley graph:

Red lines represent $s$, blue lines represent $r$. The right blue face corresponds to $r^i$ for $i \in \{0, \ldots, 3\}$. The left blue face corresponds to $s r^i$ for $i \in \{0, \ldots, 3\}$.
Now for each relation in the given presentation, we attach $2$-cells.
For $s^2 = 1$, we attach two $2$-cells to each red loop.
For $r^4 = 1$, we attach four $2$-cells to the right blue face, and another four to the left blue face.
For $rs = sr^{-1}$, we attach two $2$-cells to each face other than the two blue ones. The cells share the blue lines but go through different red lines.
The reason we attach multiple $2$-cells when a relation is a power is because we want the group to act freely on the resulting complex. If we just attach one $2$-cell, the cell will be rotated under the action but the center will stay fixed. By attaching multiple cells, we can make the action map each cell to the next one after a rotation.
