Primer on Mapping Class Groups Chapter 1: Arbitrarily short loops around Punctures Dear fellow Mathematicians,
This is the first question I ask in this forum, so please excuse any formal mistakes, which I am, of course, trying to avoid.
I am currently briefly revisiting hyperbolic geometry in the context of Thurston-Nielsen theory using the wonderful "(A) Primer on Mapping Class Groups" by Farb and Margalit.
In Chapter 1 (page 22), the authors show that any nontrivial loop on a punctured surface $S$ that can be homotoped into the neighborhood of a puncture can be made arbitrarily short. Intuitively, this is more than clear. I, however, struggle with the explanation they provide for this fact, which reads as follows:
"If a nontrivial element of $\pi_1(S)$ is represented by a loop that can be freely homotoped into the neighborhood of a puncture, then it follows that the loop can be made arbitrarily short; otherwise, we would find an embedded annulus whose length is infinite (by completeness) and where the length of each circular cross section is bounded from below, giving infinite area."
I understand that the final implication contradicts the finiteness of area of our punctured surface (which we obtain from removing points from a compact surface), but I have a hard time wrapping my head around the annulus of infinite length (this would rather work the other way round, sorry for the bad pun). Why exactly would it exist? Obviously, we assume that the length of the loop is bounded from below. But that point is probably only used in the following sentence on the circular cross sections.
I would be grateful for any advice on this short section, and I hope that this question is not a duplicate of any other question.
Best regards!
Edit: I have just found a question related to the same section in the book:
Parabolic elements correspond to punctures
The discussion here is, however, focusing on the aspect of parabolicity, whereas my question is more concerned with the embedded annulus of infinite length.
 A: Perhaps you are looking for a more rigorous explanation, but intuitively one can think of the following: Say we have a loop $\gamma$ in a closed disk $D_p$ around a puncture $p$ that goes around $p$. The topological picture is the puncture $p$, around it the loop $\gamma$, which is encircled by a circle that is the boundary of $D_p$. Drawing/thinking of the topological picture as flat one can imagine how to shrink $\gamma$ to $p$, even though we don't really have access to a notion of length yet (or rather the flat drawing is inaccurate in terms of lengths).
We have the extra hyperbolic structure on $S$, and in light of this we want to have the caricature more accurate. The argument says that even with the extra hyperbolic structure one can still shrink $\gamma$ to $p$: Otherwise we would be able to shrink $\gamma$ to $p$ in the flat drawing all the while having the length of each shrinked version of $\gamma$ bounded from below by a small positive number $\epsilon$. Thus to reconcile the shrinkage and the length being bounded from below one can think of an alternative 3D drawing of the 2D disk one of whose projections is the flat picture. The 3D drawing is that of a funnel whose pointy end points to the puncture $p$. This funnel must have infinite "height" ( = "length" of the annulus in the excerpt) for topologically we can shrink $\gamma$ to $p$ as much as we want. But since we have the lower bound $\epsilon$ the circumference of the circular cross-sections of the funnel can not decay to $0$, so that the pointy end of the funnel ought to have some positive thickness. This cannot happen since the area ought to be finite, as you declared. Thus the drawing of the disk $D_p$ that is more accurate than the flat drawing is a funnel of infinite height whose pointy end is really pointy.
