I don't understand this statement that I am trying to prove that can be found in this book (p.52):
$\langle x,y \rangle, \ x,y \in \mathrm{PSL}(2,\mathbb{C})$ reducible $\iff$ $\text{tr}[x,y] = 2$
In my version of the book, the statement looks as follows:
Either way, $[x,y] = x^{-1}y^{-1}xy $ is defined as the commutator of $x$ and $y$ and reducible is defined through all elements of $\langle x,y \rangle$ sharing a fixed point in their action on $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty \}$, given by: $$ \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{C}), \qquad \gamma\cdot z = \frac{az+c}{cz+d}$$
So far, I have been unable to prove either direction. My current approach, uses $\mathrm{tr}([x,y])= 2 \iff [x,y]$ is parabolic, meaning it has only a single fixed point. First, I am considering the transformations that fix $0, \infty$, which allows me to think in a simpler geometric way about $x,y, [x,y]$, but I have been unable to get the desired result.