Books and references for Möbius transformation, hyperbolic Riemann surface and covering transformation I am reading the book Computational Conformal Geometry by Xianfeng David Gu and Shing-Tung Yau. There is a part in the book which I don't understand and I would like to ask for books and references explaining the material in detail.
On page 128, the authors classified the Möbius transformations of the Riemann sphere (other than the identity map) into four types. A Möbius transformation is

*

*parabolic if it is has a unique fixed point and hence is conjugate to the map $z\mapsto z+1$.
Now any other Möbius transformation (other than the identity) has two distinct fixed points, and are further classified. Say $\alpha$ is such a map with two fixed points. By the existence of a Möbius transformation $\gamma$ that sends three distinct points $z_1,z_2,z_3$ on the sphere to another three points $w_1,w_2,w_3$ on the sphere (i.e. $\gamma(z_i)=w_i$ for each $i$), select a Möbius transformation $\gamma$ that sends the two fixed points to $0$ and $\infty$ respectively. Then $\gamma\circ\alpha\circ\gamma^{-1}$ has $0$ and $\infty$ as its fixed points, so that it must be of the form $z\mapsto kz$. Write $k=re^{i\theta}$, $r\gt0$, $\theta\in\Bbb R$. Then $\alpha$ is said to be


*elliptic if $r=1$ and $\theta\not=0$;

*hyperbolic if $r\not=1$ and $\theta=0$;

*loxodromic if $r\not=1$ and $\theta\not=0$.

Then the authors made the following claims, which I quote exactly.

If $\alpha$ is a parabolic or a hyperbolic transformation preserving $\Delta$, then its fixed points are on $\partial\Delta$. If $\alpha$ is an elliptic transformation preserving $\Delta$, then one of its fixed points is inside $\Delta$, the other is outside $\Delta$.
Theorem 6.40. Suppose $M$ is a Riemann surface covered by $\Delta$, the covering transformation group is $G$. Then the nontrivial element in $G$ is either hyperbolic or parabolic. If $M$ is compact, then all non-trivial elements are hyperbolic.

The claim that a map with a unique fixed point is conjugate to $z\mapsto z+1$ and the claim before Theorem 6.40 are not proved in the book. Theorem 6.40 is "proved" in the book but the proof is incomprehensible to me. They also seem to claim implicitly that two Möbius transformations $\alpha,\beta$ of different types cannot be conjugate to each other.
Which books and references can I find detailed proofs of the above claims and theorem?
Addendum: $\Delta$ is the unit disk.
Edit: I found an online source on these things, but the proofs are missing: http://www.math.tifr.res.in/~pablo/download/fuchsian.pdf
 A: The claim about parabolic transformation being conjugate to $z\mapsto z+1$ is proved as below. For any parabolic transformation $\alpha$, let $p$ be the fixed point of $\alpha$ and consider a Möbius transformation $\gamma$ that sends $p$ to $\infty$. Then the fixed point of $\gamma\circ\alpha\circ\gamma^{-1}$ is $\infty$. This condition forces $\gamma\circ\alpha\circ\gamma^{-1}$ to be of the form $z\mapsto z+b$ for a constant $b\in\Bbb C\setminus\{0\}$. To see this, write $\gamma\circ\alpha\circ\gamma^{-1}(z)=\frac{az+b}{cz+d}$, and put $z=\infty$ to see $\frac{a}{c}=\infty$, giving $c=0$. Without loss of generality, we may put $d=1$. So now $\gamma\circ\alpha\circ\gamma^{-1}$ is of the form $z\mapsto az+b$. If $a\not=1$, we can solve the equation $az+b=z$ to find a fixed point other than $\infty$, giving a contradiction. So $a=1$.
To see that $z\mapsto z+b$ is conjugate to $z\mapsto z+1$, write down the matrix representation of $z\mapsto z+b$, i.e. $$\begin{bmatrix}
1 & b\\
0 & 1
\end{bmatrix}.$$ This matrix has Jordan canonical form $$\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}$$ which represents $z\mapsto z+1$. 
The claim about the fixed points of parabolic, elliptic and hyperbolic transformations that preserves a disk is proved in the book The Geometry of Discrete Groups by Alan F. Beardon, by page 93. Moreover, it is stated and proved that if there is a loxodromic transformation in a subgroup $G$ of group of all Möbius transformations, then there is no $G$-invariant disk. 
Theorem 6.40 is proved in the book Riemann Surfaces, Second Edition, by Hershel M. Farkas and Irwin Kra, by page 232, as Corollary 2 in this book. Though the statement is stated as "if $G$ is a Fuchsian group which is the covering group of a compact Riemann surface of genus greater than or equal to $2$...", it is essentially the same because saying a compact Riemann surface to have genus $\ge 2$ is the same as saying the universal covering is the unit disk.
