Since the set of automorphisms of the disk in the complex plane is a group, we could ask the question of whether there exists a topological space such that the fundamental group of that space is a group isomorphic to the automorphisms of the disk?
Moreover I have a secondary question which might sound dumb: can an automorphism of the disk ($D$ in $\mathbb{C}$) be viewed as a path connecting two points of $D$? if not is it because an element of Aut($D$) map a point of $D$ to a point of $D$ in a "discontinuous way", like "a jump from one point to another" (because the image of such a map is a singleton) ?