I am reading this paper: https://arxiv.org/pdf/math/0611409.pdf
On page 41 in the section Plumbing a cylinder it is writen:
by (4.1), a simple pole of an Abelian differential has a neighborhood which is isometric to a half infinite cylinder.
I do not understand that part. The following is the preceding part where is shown equation (4.1)
Translation surfaces. Let $X$ be a Riemann surface with a nonzero Abelian differential ω, and let Z(ω) be the discrete set of zeros of ω. Every point of X which is not a zero of ω has a neighborhood U and a conformal map $φ: U → \mathbb{C}$ such that $ω|_U = φ^∗(dz)$. The conformal map φ can be defined explicitly by $φ(z) = \int_{z_0}^z ω$ for a choice of base point $z_0$. These coordinates φ are unique up to translation by a constant. We can also put ω in a standard form in the neighborhood of a zero or a pole at $z_0$.There is a neighborhood U of $z_0$ and conformal map $φ_0 :(U,z_0) \to(\mathbb{C},0)$ such that
$ω|_U =φ^∗_0(z^ndz)$, if $z_0$ is not a simple pole; or $φ^∗_0(a dz/z)$, if $z_0$ is a simple pole with nonzero residue $a$. This is equation (4.1).
I will appreciate any suggestion on this. Thank you.