Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations

Question

Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$ Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on $\mathbb{H}$ by fractional linear transformations.

The notes I had on the subjet prove that for $z,z'\in \mathbb{H}$ with $d(z,z')=r$, then the hyperbolic sphere is given by $$S(z,r)=\text{stab}_{\text{SL}_2(\mathbb{R})}(z).z'.$$

However, the proof contains a unfixable mistake and I have not been able to find another one (nor to find one myself). Is there any book where I could find that, or would you have any good idea to do that ?

Note that the inclusion $S(z,r)\supset\text{stab}_{\text{SL}_2(\mathbb{R})}(z).z'$ is obvious since $\text{SL}_2(\mathbb{R})$ acts by isometries.

Answer 1

I like to switch between three models of $\mathbb H$ depending on the situation:

• half-space model when there is a distinguished boundary point
• disk model when there is a distinguished interior point
• infinite strip model when there is a distinguished geodesic

So in this case I would switch to the disk model and map $z$ to the center of the disk. Then the orbit of $z'$ under rotations about the origin is clearly $S(z,r)$.