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Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, by van-Kampen theorem, this group is an product of two 2-free groups amalgamated by Z, but I really need to literally see that.

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  • $\begingroup$ @EricWofsey Ah, I didn't know the phrase "$n$-torus" has two meanings. $\endgroup$ – arctic tern Jun 21 '16 at 4:51
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    $\begingroup$ Maybe it's better to refer to the genus of this thing? Do you know the usual presentation of it as $\langle a_1,b_1,a_2,b_2 : [a_1,b_1][a_2,b_2]\rangle$? $\endgroup$ – Hoot Jun 21 '16 at 4:57
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    $\begingroup$ What exactly are you looking for from something as awkward as the Cayley graph that description of $\pi_1 (\Sigma_2)$ as an amalgamated free product wouldn't give you? The usual presentation of $\pi_1(\Sigma_2)$ gives a convenient generating set. $\endgroup$ – anomaly Jun 21 '16 at 5:06
  • $\begingroup$ I just need to see what happens with the Cayley graphs of amalgamated products. This is just an easy example. $\endgroup$ – Lucas Henrique Jun 21 '16 at 5:10
  • $\begingroup$ I still don't know what you're trying to get out of this, but you might find something useful in Serre's "Trees." $\endgroup$ – anomaly Jun 21 '16 at 5:25
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It looks like this:

A closed orientable surface of genus $2$ admits a hyperbolic metric, so corresponds to some quotient of the hyperbolic plane $\mathbb{H}$. The action of the fundamental group on $\mathbb{H}$ can be used to draw a Cayley graph in $\mathbb{H}$. This is what this picture is depicting, in the Poincaré disk model of $\mathbb{H}$.

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    $\begingroup$ That's almost exactly what I'm looking for. My problem with this picture is that becomes too small really fast, so it is not possible to see so much things on that. Do you have another picture? Perhaps on the half-plane model or on the klein disk model. $\endgroup$ – Lucas Henrique Jun 21 '16 at 6:14
  • $\begingroup$ This is an amazing picture. How did you generate it? $\endgroup$ – NWMT Apr 11 at 12:40
  • $\begingroup$ @NWMT: Google image search! $\endgroup$ – Qiaochu Yuan Apr 11 at 18:58

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