The complex unit (open or closed) disk as a group I tried to search some literature on this specific argument, but didn't find anything really up to the point and well written. The questions I'm interested in are: 
- what are the (relevant) operations which form a group with the unit disk?
- are these groups somehow important in geometry or in algebra? 
Any reference is wellcome. 
 A: Look at the subgroup of Mobius transformations that sends the unit circle to the unit circle (and the inside of the unit circle to the inside of the unit circle).
Now let $z$ in the open unit circle represent a Mobius transformation that takes $0$ to $z$. This becomes a group. Well, almost. Because the complex number $0$ can represent any rotation, and similarily, an arbitrary $z$ can represent a transformation $f$ but also $f$ preceeded by any rotation, we need to narrow it down further. So what we need is the subgroup of Mobius transformations that preserve the unit disc and fixes a single point on the unit circle, like, for instance, $1$.
A: The map
$$f:\quad {\mathbb C}\to D,\qquad z\mapsto w:={z\over 1+|z|}$$
with inverse
$$f^{-1}:\quad D\to{\mathbb C},\qquad w\mapsto z:={w\over 1+|w|}$$
is a homeomorphism from ${\mathbb C}\sim{\mathbb R}^2$ to the open unit disk $D\subset{\mathbb C}$. Now transport the addition in ${\mathbb C}$ to $D$ using $f$. This turns $D$ into an additive group. In terms of a formula: For $u$, $v\in D$ define
$$u\oplus v:=f\bigl(f^{-1}(u)+f^{-1}(v)\bigr)\ .$$
