This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta and Seshadri which can be found here:
http://repository.ias.ac.in/20407/1/305.pdf
On page 207 (the 3rd page of the PDF), they introduce the following notation:
My question is: What is the definition of a parabolic cusp? I would also like to know the motivation behind the name. A reference would be nice, as well. Thank you.
I think we have a winner, The Geometry of Discrete Groups by A. F. Beardon, Springer-Verlag 1983.
From the second link in comment, "Let $\Gamma$ be a discrete subgroup of $\operatorname{PSL}(2,\Bbb{R})$, the group of fractional transformations $z\mapsto (az + b)/(cz + d)$; $a,b,c,d\in\Bbb{R}$, $ad - bc = 1.$ For any $z$ in the extended complex plane and any sequence of distinct elements $\gamma_i$ of $\Gamma$ a cluster point of $\{\gamma_i z\}$ is called a limit point of $\Gamma$. If there are 0, 1 or 2 limit points, $\Gamma$ is conjugate to a group of motions of the plane. Otherwise there are infinitely many limit points and $\Gamma$ is called a Fuchsoid group. A Fuchsoid group is a Fuchsian group if it is finitely generated. For a real point $x\in\Bbb{R}\cup\{\infty\}$ let $\Gamma_x$ be the stabilizer in $\Gamma$ of $x$. The point $x$ is called a cusp or parabolic cusp if $\Gamma_x$ is a free cyclic group generated by a parabolic transformation $\neq\{1\}$ (cf. Fractional-linear mapping). The cusps of $\Gamma$ are represented by vertices of a fundamental polygon on the real axis."
I also put the words
parabolic cusp fuchsian group
in Giggle, found a pdf for a very nice 2002 paper by Long and Reid, which expands on Thurston's MO answer.