Denote the upper half space by $\mathcal{H}\_{3}=\Bbb{C}\times (0,\infty)$. A point $P \in \mathcal{H}\_{3}$ is given as, $P=(z, t)=(x, y, t)=z+t j$ where $z=x+i y$ and $j=(0,0,1) .$ The group $P S L\_{2}(\mathbb{C})$ has a natural action on $\mathcal{H}\_{3} .$ Let $M=\left(\begin{array}{c}\alpha & \beta \\ \gamma & \delta\end{array}\right)$
and $P=z+t j$ a point in $\mathcal{H}\_{3}$. Then, $P S L\_{2}(\mathbb{C})$ acts on $\mathcal{H}\_{3}$ via linear fractional transformation as follows
$$
M \times P \rightarrow \frac{\alpha P+\beta}{\gamma P+\delta}
$$
More explicitly we have $M(z+t j)=z^{\*}+t^{\*} j \in \mathcal{H}\_{3}$ where
$$
\begin{array}{l}
z^{\*}=\frac{(\alpha z+\beta)(\bar{\gamma} \bar{z}+\bar{\delta})+\alpha \bar{\gamma} t^{2}}{|\gamma z+\delta|^{2}+|\gamma|^{2} t^{2}} \\
t^{\*}=\frac{t}{|\gamma z+\delta|^{2}+|\gamma|^{2} t^{2}}
\end{array}
$$


I want to prove the following three statements: Let $M\in PSL\_2(\Bbb{C})$.


* $M$ is it has a fixed point in $\mathcal{H}\_{3}$ $\Longleftrightarrow \operatorname{Tr}(M) \in(-2,2)$
* $M$ has no fixed point in $\mathcal{H}\_{3}$ and a single fixed point on $\partial \mathcal{H}\_3, \Longleftrightarrow \operatorname{Tr}(M)=\pm 2,$
* $M$ has no fixed point in $\mathcal{H}\_3$ and exactly 2 fixed points on $\partial \mathcal{H}\_3$. $\Longleftrightarrow \operatorname{Tr}(M) \notin[-2,2]$ (and hyperbolic $\Longleftrightarrow \operatorname{Tr}(M) \in \mathbb{R} \backslash[-2,2])$.


My attempt:
If $M$ fixes a point $P$ then we have the following equations


$$\frac{(\alpha z+\beta)(\bar{\gamma} \bar{z}+\bar{\delta})+\alpha \bar{\gamma} t^{2}}{|\gamma z+\delta|^{2}+|\gamma|^{2} t^{2}}=z$$ and $$\frac{t}{|\gamma z+\delta|^{2}+|\gamma|^{2} t^{2}}=t$$


So this gives $|\gamma z+\delta|^{2}+|\gamma|^{2} t^{2}=1$ and $(\alpha z+\beta)(\bar{\gamma} \bar{z}+\bar{\delta})+\alpha \bar{\gamma} t^{2}=z$. After these two equations, I am not able to solve $z$ and $t$. I want to see relations between the trace and fixed points. I would be very glad if someone is able to relate the trace of $M$ to the fixed points of $M$ from my equations. Thanks a lot.



Classification of isometries of hyperbolic 3-space
Let $M\in\mathrm{PSL}\_2(\mathbb{C})$ be distinct from $\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$. If $M$ has distinct eigenvalues, then it is conjugate within $\mathrm{PSL}\_2(\mathbb{C})$ to a diagonal matrix. If $M$ has equal eigenvalues, then it is conjugate within $\mathrm{PSL}\_2(\mathbb{C})$ to $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$.


The three statements in the original post are invariant under conjugation, so it suffices to verify them for diagonal matrices and $\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$. Which is straightforward.


