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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?

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    $\begingroup$ Neither of those two examples has torsion. All of their elements apart from the identity have infinite order. $\endgroup$
    – Derek Holt
    Commented Apr 22, 2014 at 20:39
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    $\begingroup$ There is a Cayley graph for any group and any generating set. en.wikipedia.org/wiki/Cayley_graph $\endgroup$
    – Lee Mosher
    Commented Apr 22, 2014 at 21:48
  • $\begingroup$ @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? $\endgroup$
    – BIS HD
    Commented Apr 23, 2014 at 19:50
  • $\begingroup$ @DerekHolt You're right, I'm sorry. I removed the "like the two groups above" part. $\endgroup$
    – BIS HD
    Commented Apr 23, 2014 at 19:52
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    $\begingroup$ @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. $\endgroup$
    – YCor
    Commented Apr 30, 2014 at 22:34

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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.

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  • $\begingroup$ 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. $\endgroup$
    – BIS HD
    Commented Apr 23, 2014 at 19:53
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    $\begingroup$ 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. $\endgroup$
    – Moishe Kohan
    Commented Apr 23, 2014 at 19:57

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