Symmetries of the Upper Half Plane and the Unit Disk

In this post, we will show:

Step 1: we show that there exists a diffeomorphism between the upper half plane and the unit disk. This then tells us that the group of symmetries of the upper half plane is isomorphic to the group of symmetries of the unit disk.

Step 2: We show that is a subgroup of the symmetries of the upper half plane, and .

Step 3: We show that the group of symmetries of the unit disk is $latex PSU(1,1).

And then, it follows that:

Step 1:

Step 2.a:

Step 2.b:

It is then easy to see that the map we have constructed from is a bijection and preserves the group structure.

Step 3: PSU(1, 1) is the group of symmetries of the unit disk.