Isometry of $\mathbb H^2$ determined by its action on $\partial H^2$.

Question

On pg. 282 of Farb and Margalit's A Primer on Mapping Class Groups, the following is mentioned:

An isometry of $\mathbb H^2$ is determined by its action on $\partial \mathbb H^2$.

I think here $\partial \mathbb H^2$ refers to the Gromov boundary of $\mathbb H^2$. So it is clear to me that $\text{Isom}(\mathbb H^2)$ acts on $\partial \mathbb H^2$ in a natural way. But can somebody please explain as to how an isometry of the upper half plane is determined by its behavior on the boundary alone. Or give a reference? Thanks.

Answer 1

Isometries of the upper half plane are linear fractional transformations which preserve the real line. A linear fractional transformation is determined by three points, so just pick your favorite three points on the boundary and they'll tell you what your isometry is.

Answer 2

This is just Moebius transformations with real coefficients, acting on the upper half plane. To reverse orientation, then map to $- \bar{z}$