Is there a cayley graph for the Klein bottle?

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?

• Neither of those two examples has torsion. All of their elements apart from the identity have infinite order. – Derek Holt Apr 22 '14 at 20:39
• There is a Cayley graph for any group and any generating set. en.wikipedia.org/wiki/Cayley_graph – Lee Mosher Apr 22 '14 at 21:48
• @LeeMosher Ok, I was aware about the caley graph of any generating set, but how to model relations in a Caley graph? And why the downvote? – BIS HD Apr 23 '14 at 19:50
• @DerekHolt You're right, I'm sorry. I removed the "like the two groups above" part. – BIS HD Apr 23 '14 at 19:52
• @BISHD to model relators of a presentation, you consider the so-called Cayley 2-complex: for every relator, say of length $k$, you fill the given loop by a $k$-gon, as well as all its translate. You can do this for any set of relators (i.e. subset of the kernel $F_S\to G$, where $S$ is the generating subset); the resulting complex is simply connected iff the given relator generate the kernel as a normal subgroup, i.e. define a presentation of the group. – YCor Apr 30 '14 at 22:34

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.

• Thank you for your answer. Is the standard planar square grid a Cayley graph in the usual sense? I'm not too sure about that. – BIS HD Apr 23 '14 at 19:53
• Yes, there is only one sense. As for the proof, try to find an action of Klein bottle group on the grid such that the quotient graph has 1 vertex and 2 edges. If you cannot, let me know. – Moishe Kohan Apr 23 '14 at 19:57