Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed. If we want to parametrise annuli with punctures at one boundary, give the punctures location up to a rotation around the origin.
If we represent an annulus as the quotient of the Poincare' disk by an hyperbolic element $\alpha$ of appropriate modulus $D(\alpha) := \inf_{z \in D} d(z,\alpha(z)),$ we can see that in the limit $D(\alpha) \to \infty$ (the annulus get "pinched") we get a Poincare' disk with two extra markings. We can think of this as the result of normalising the double point of the pinched annulus.
We can imagine the pinching happens because the original annulus was obtained removing a circle $C$ from the Poincare' disk $D$ and that now $C$ is tending to an horocycle, while mantaining a non-zero euclidean radius.
I wonder if we can think of the pinched annulus as what we obtain if we cut an horocycle from $D$, and if we can parametrise marked pinched annuli with the position of the markings on $\partial D$ up to parabolic transformation with fixed point $1 \in \partial D$: this seems the limit of the parametrisation of marked annuli in the limit when the rotations around $0$ become "ideal" rotations around $1$.
