I have the following Cayley graph: $$ ...\cdot \leftrightarrow \cdot \leftrightsquigarrow \cdot \leftrightarrow \cdot \leftrightsquigarrow \cdot \leftrightarrow ...$$

Here are my thoughts: we have two generators $a,b$. And $a^2=b^2=e$ since we have arrows going out and back from $e$ to $a$ and $b$. So this group is the group of words $ababa...aba$ which can start and end with either $a$ or $b$. Is it the same as a free group with two relations: $\langle a, b\ |\ a^2, b^2\rangle$? Is my interpretation correct?

  • 2
    $\begingroup$ I'm really not keen on 'a free group with two relations'. There are relations, so it's not free. You can call it a two-relator group. $\endgroup$ – Tara B Apr 18 '12 at 11:14
  • $\begingroup$ But the terminology is almost correct: this is a free product of the two order 2 groups $\langle a \, | \, a^2\rangle \approx \mathbb{Z}/2\mathbb{Z}$ and $\langle b \, | \, b^2\rangle\approx \mathbb{Z}/2\mathbb{Z}$. $\endgroup$ – Lee Mosher Nov 12 '15 at 15:49

Yes, your interpretation is correct. There are no further relations since there are no loops in the Cayley graph except those deducible from the relations $a^2 = b^2 = e$.

The group $\langle a, b \mid a^2, b^2 \rangle$ is known as the infinite dihedral group.


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