Triply transitive action on upper half plane

complex-analysishyperbolic-geometrysolution-verification

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.

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Can I transform this triangle to the following triangle via an element of $PSL(2, \mathbb{R})$?

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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. :)

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