Is there a cayley graph for the Klein bottle?

Question

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to

$$\langle a, b \mid aba^{-1}b^{-1} \rangle$$

(the free abelian group on two generators)

Analogously, we can compute the fundamental group $\pi_1(K^2)$ of the Klein bottle, that is isomorphic to

$$\langle a,b \mid abab^{-1} \rangle $$

Now, when starting to consider Cayley graph, I noticed that there are constructions to picture a free group on two generators (non-abelian). Are there similar constructions in order to picture groups with torsion?

For the cyclic group $\Bbb Z_n$, one can consider the $n$-cycle graph.

Is there a consequent way of constructing graph representations of groups with relations?

Answer 1

You can use the standard planar square grid as a Cayley graph for $\pi_1(K^2)$: It is the same as the one for the torus, just the group action is different.