For any matrix

$$A=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in S L(2, \mathbb{R})$$

the corresponding Möbius transformation is

$$z \mapsto A z=\frac{a z+b}{c z+d},$$

which acts on the upper half-plane $\mathbb{H}$, equipped with the hyperbolic metric $\rho$.

(a) Assuming that $|\operatorname{tr} A|>2$, prove that $A$ is conjugate in $S L(2, \mathbb{R})$ to a diagonal matrix $B$. Determine the relationship between $|\operatorname{tr} A|$ and $\rho(i, B i)$.

(b) For a diagonal matrix $B$ with $|\operatorname{tr} B|>2$, prove that

$$\rho(x, B x)>\rho(i, B i)$$

for all $x \in \mathbb{H}$ not on the imaginary axis.

(c) Assume now that $|\operatorname{tr} A|<2$. Prove that $A$ fixes a point in $\mathbb{H}$.

(d) Give an example of a matrix $A$ in $S L(2, \mathbb{R})$ that does not preserve any point or hyperbolic line in $\mathbb{H}$. Justify your answer.