I have some problems with the proof(you can see the paper here)
In Section 3(Page 7) the article writes
Let $\mathbb{R}^2/\Lambda$ be a torus of area at least 1.
And in the next page(Page 8), the article writes
Then move the cluster to the cylinder $\mathbb{R}^2/\mathbb{Z}v$, where $v$ is the translation along the length of the diameter.
Here, I can't understand why $\mathbb{R}^2/\mathbb{Z}v$ is a cylinder and why $\mathbb{R}^2/\Lambda$ is a torus. More basically, what does $/$ mean here?
$\Bbb R^2/\Lambda$ is the quotient group of $\Bbb R^2$ by the subgroup $\Lambda$. It is given the quotient topology induced from $\Bbb R^2$. If $\Lambda$ is a rank two lattice in $\Bbb R^2$, then $\Bbb R^2/\Lambda$ is homeomorphic to the torus.
As an example if $\Lambda=\Bbb Z^2$, then $\Bbb R^2/\Lambda=\Bbb R^2/\Bbb Z^2 =(\Bbb R/\Bbb Z)^2$, the product of two circles $\Bbb R/\Bbb Z$.