I am trying to learn decorated Teichmüller theory and reading Penner's "Decorated Teichmüller Theory" book. In Chapter 2 in "Punctured Surfaces" section I got confused.
Let $\mathbb{D}$ denote the Poincare disc model of the hyperbolic surface. Let $\Gamma$ denote Fuchsian group. He wrote:
"Suppose that $x \in S_{\infty}$ is the fixed point of some parabolic transformation in $\Gamma$ acting on $\mathbb{D}$. A horocycle in the hyperbolic plane centered at $x$ projects into the surface $F=\mathbb{D}/\Gamma$ to a closed curve in $F$ , and we refer to such a curve as a horocycle in $F$ , which is possibly immersed but not embedded; we say that the horocycle is centered at the puncture of $F$ corresponding to $x$."
I don't understand why a horocycle in $\mathbb{D}$ becomes a closed curve in $\mathbb{D}/\Gamma$? why can't it intersect itself?
Thank you!