I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a
"geometrically finite hyperbolic surface of infinite volume"
is mentioned frequently and I am struggeling to locate a precise definition for it (presumably because it is a relatively basic concept that one should know about).
If someone could add some definition to the above words (or refer to an appropriate source where I could read up on this) that would be very helpful, many thanks !
A google search for geometrically finite manifold gave this as the first hit. Briefly, a hyperbolic manifold is geometrically finite if it has finitely many connected components, each of which is the quotient of hyperbolic space by a geometrically finite group of isometries, this latter notion being defined in terms of the existence of a certain nice fundamental domain. So a connected hyperbolic manifold is geometrically finite if it can be obtained by identifying the edges of a fundamental domain in hyperbolic space in a certain (nice) way.
It would seem to me that the right way to get intuition for this notion would be to explore some simple examples of discrete subgroups of $\mathrm{SL}(2,\mathbb R)$ and their fundamental domains in $\mathbb H^2$. If you haven't considered it yet, $\mathrm{SL}(2,\mathbb Z)$ is a natural group to start with (and its fundamental domain is particularly famous and easy to learn about), and you can then move on to some of its subgroups (such as the one I consider below, generated by the matrix $\begin{pmatrix} 1 &1 \\ 0 & 1\end{pmatrix}$).
As for infinite volume, it means exactly what it says: you have a hyperbolic surface, i.e. a surface equipped with a Riemannian metric of constant negative (Gaussian) curvature, and the intergal of the corresponding volume form over the surface is infinite.
As noted in the linked wikipedia entry, in dimension two, any quotient of the hyperbolic plane by a finitely generated discrete group of isometries gives a geometrically finite hyperbolic surface. So one example of the kind of surface you are looking at would be the complex upper half-plane (one of the standard models for the hyperbolic plane) modulo the symmetery $z \mapsto z+1$. (This is topologically a cylinder, but it is equipped not with the usual flat metric that we think of a cylinder as carrying, but rather with a hyperbolic metric.)