# Cayley graph of the fundamental group of the 2-torus

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.

• @EricWofsey Ah, I didn't know the phrase "$n$-torus" has two meanings. Jun 21, 2016 at 4:51
• Maybe it's better to refer to the genus of this thing? Do you know the usual presentation of it as $\langle a_1,b_1,a_2,b_2 : [a_1,b_1][a_2,b_2]\rangle$?
– Hoot
Jun 21, 2016 at 4:57
• 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. Jun 21, 2016 at 5:06
• I just need to see what happens with the Cayley graphs of amalgamated products. This is just an easy example. Jun 21, 2016 at 5:10
• I still don't know what you're trying to get out of this, but you might find something useful in Serre's "Trees." Jun 21, 2016 at 5:25

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 $\mathbb{H}$.