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.
2025-01-13 05:29:48.1736746188
Cayley graph of the fundamental group of the 2-torus
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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 $\mathbb{H}$.