I've been working on the following problem, and although I've made some progress, I don't know how to finish. Throughout the exercise, we are working in the Poincaré disc.
Consider the points $p=(-1/3,0)$ and $q==(1/3,2/3)$, and let $T$ be the hyperbolic translation such that $T(p)=q$. Let $R$ be the hyperbolic rotation around $p$ in $\pi/2$. Determine if $TR$ is a hyperbolic translation, rotation or parallel displacement.
Progress so far: The translation $T$ takes a point and sends it to another point that lies on the Euclidean circle that passes through the point and the points $(\frac{-12\sqrt{13}-15}{61},\frac{-10\sqrt{13}+18}{61})$ and $(\frac{12\sqrt{13}-15}{61},\frac{10\sqrt{13}+18}{61})$. I did this by a direct computation, by finding the hyperbolic line that goes through $p$ and $q$ and seeing where it intersects the unit circle. Next, the rotation around $p$ in $\pi/2$ is the composition of two reflections, one with respect to the line $y=0$ and the other one with respect to the circle $(x+5/3)^2+(y-4/3)^2=32/9$. I also did this by direct computation.
Now, in order to see what $TR$ is, I believe I would want to be looking for fixed points, for example. However, the only way I can think of doing this is by explicitly writing out the formula for $TR$. This doesn't seem elegant or desirable. Can anyone help me?
$^*$I recently posted this accidentally using someone else's account that also uses this computer; I deleted that question and am asking it again.
Ok, I've been able to work it out. Here's the answer (I've omitted many of the explicit calculations since they are fairly easy and a bit tedious to write out):
The hyperbolic line that goes through $p$ and $q$ has equation $N:(x+5/3)^2+(y-2)^2=\frac{52}{9}$. The hyperbolic line that goes through $p$ and is orthogonal to this circle has equation $M:(x+5/3)^2+(y+8/9)^2=208/81$, and by an explicit calculation we find that the translation $T$ can be obtained by reflecting around the circle $L:(x+5)^2+(y+14/3)^2=25+14^2/9-1$ and then reflecting around $M$.
Now to calculate the rotation around $p$, let $K$ be the circle $(x+5/3)^2+(y+4/15)^2=25/9+16/15^2-1$. This circle passes through $p$ and intersects $M$ in an angle of $\pi/4$. Therefore, rotation around $p$ in $\pi/4$ can be calculated by reflecting first around $K$ and then $M$. Therefore, $TR$ can be obtained by reflecting first around $K$, then $M$, then $M$ and finally $L$. In particular, this is equal to reflecting around $K$ and then $L$. Since $K$ and $L$ are disjoint, we obtain that $TR$ is a translation.
I hope this makes sense.