I'm trying to understand a proof of a theorem from hyperbolic geometry, with very much detail. So I will post here the complete proof, and mark some things in bold and purple, in the hope that I get a "better" explained or more like in laymans, mathematical terms. Or maybe some insight about the important thing to understand with this proof. I write the theorems and propositions below the proof in case they are needed. I hope this is not too much to ask.

The theorem is:

Any isometry from the hyperbolic plane, $\mathbb{H}^2$ is an element of $PSL(2, \mathbb{R}),$ or it is of the form $$ z \rightarrow \frac{a(-\overline{z} ) + b}{c(-\overline{z} ) + d},$$ where $a,b,c,d \in \mathbb{R}, ad-bc = 1.$

Proof. Let $\varphi$ be an isometry of $\mathbb{H}^2,$ with the Hyperbolic Metric.

Using theorem 1, 2, and the proposition ( and if it is necessary , applying a homotethy and the function $ z \rightarrow \frac{-1}{z}$ you can suppose, without loss of generality, that there exists a $\overline{T} \in PSL(2, \mathbb{R}), $ such that the function $ \overline{T}\varphi$ fixes punctually the positive imaginary axis.

Now, let $z \in \mathbb{H}^2 , z = x + iy$ and $ \overline{T}\varphi = a + ib.$ As a consequence of $cos \rho(z,w) = 1 + \frac{|z - w|^2}{2ImzImw},$ we then have that for all $\color{purple}{t \in \mathbb{R}, t>0,}$ $\color{purple}{\frac{|it - \overline{T}\varphi^2|}{2Im(it)Im(\overline{T}\varphi(z))} } $ (why is the 1 missing?) $= \frac{|it - z^2|}{2Im(it)Im(z)}.$ Therefore $$ \frac{a^2 + (b-t)^2}{b} = \frac{x^2 + (y - t)^2}{y}$$ $$\Rightarrow y(a^2 + (b-t)^2) = b (x^2 + (y - t)^2).$$ As $\color{purple}{t \to \infty },$ we obtain $\color{purple}{b = y}$ and $\color{purple}{a = \pm x.}$ As isometries are evidently continuous, then, the points in the first quadrant, where $\color{purple}{\overline{T}\varphi}$ is the function $ \color{purple}{z \to - \overline{z}} $ (or the identity), is an open and closed set (this argument applies to the second quadrant). Then, by connectedness, and because every isometry is injective we get $ \color{purple}{\overline{T}\varphi = Id} $ or $\color{purple}{\overline{T}\varphi(z) = -\overline{z}}$ (Why am I trying to prove these last statements?) End of the proof.

theorem 1. The group $PSL(2, \mathbb{R})$ acts as a group of isometries in $\mathbb{H}^2$ with the hyperbolic metric.

Theorem 2. $\rho(z,v) = \rho(z,w) + \rho(w,v) $ if and only if $w \in [z,v],$ for $z,w,v$ three distinct points in $\mathbb{H}^2$

Proposition: The group $PSL(2, \mathbb{R})$ acts transitively in the family of circles orthogonal to the real axis.


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