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I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und J├╝rgen Jost. Right now I'm looking at an exercise (12.5) under the description hyperbolic group.

Here's the exercise:

1) Show that for every three real numbers $x_0 < x_1 < x_2$ there exists exactly one reel rational function $ f(x)=\frac{ax+b}{cx+d}$ with $f(0)=x_0$, $f(1)=x_1$, $f(\infty)=x_\infty$ and $f$ is supposed to be invertible.( where $f(\infty)=\lim \frac{ax+b}{cx+d}=\frac{a}{c}$) Extended to $\mathbb C$ f maps the upper halfplane $\mathcal H$ onto itself.

2) Conclude that to two reel triples of numbers $x_1 < x_2 < x_3 $ and $y_1 < y_2 < y_3 $ there exists a rational function $f$ with $f(x_i)=y_i$ for $i=\{1,2,3\}$ and $f|_H$ is an hyperbolic isometry.

I have absolutely no idea where or how to begin to solve this exercise.

What i do know is that the hyperbolic plane and the $S^2\subset \mathbb R^3$-sphere both share the same orthogonal group $O(3)$. And the hyperbolic plane $\mathcal H\subset \mathbb C$ has a three-parameter group $G=SL(2,\mathbb R)$ of all reel $2x2$matrices.

I would be glad for any help regarding the solution of this exercise.

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