I am calculating the area of a triangle in the upper half plane. Consider the following triangle in the upper half plane with the Poincare metric.
Can I transform this triangle to the following triangle via an element of $PSL(2, \mathbb{R})$?
I have the above doubt. I think I can do this because $PSL(2, \mathbb{R})$ triple transitively on boundary of the upper half plane, which is $\Bbb R \cup \{ \infty \}$.
Am I correct?
Best Answer
Yes, you're right, that you can map any hyperbolic triangle with all vertices on $\mathbb R\cup \{i\infty\}$ to any other, by the action of $SL_2(\mathbb R)$.
For me, it's clearest to do it in steps, and to map to the hyperbolic triangle with vertices $0,1,i\infty$. Namely, given such a triangle, map any vertex to $i\infty$. Then translate so that the left-most vertex on the real line is moved to $0$. The dilate to move the right-most vertex to $1$.
So, yes, semi-amazingly, all their hyperbolic areas are the same. :)