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I'm currently reading a paper by Jason Jeffers titled "Lost Theorems of Geometry" published in The American Mathematical Monthly vol. 107, no.9 (Nov. 2000) pages 800-812.

The authors proves three theorems there and I have problems understanding the proof of this one:

Suppose that \(\displaystyle f: \mathbb{H}^2 \rightarrow \mathbb{H}^2\) is a bijection which preserves geodesics. Then \(\displaystyle f\) is an isometry of \(\displaystyle \mathbb{H}^2 \).

\(\displaystyle \mathbb{H}^n \) - hyperbolic \(\displaystyle n\)-space

The author says (quote) :

"The essential idea is to show first that the map is continuous, and then that it is identical to an isometry on a dense subset of points. Actually, all of this takes place on the boundary of \(\displaystyle \mathbb{H}^2 \)., so our first task is to show that it makes sense to refer to \(\displaystyle f\) on the boundary".

Then we prove that it makes sense to think of \(\displaystyle f\) as a map on \(\displaystyle \mathbb{H}^2 \) and its boundary by proving a lemma which says that if two geodesics in \(\displaystyle \mathbb{H}^2 \) have a common endpoint on the boundary, then their images also have a common endpoint on the boundary. We only assume that \(\displaystyle f\) is a bijection which preserves geodesics.

Up to this point the only thing I don't fully understand is why is it so important to know that we can think of \(\displaystyle f\) as a map on the hyperbolic space plus its boundary. What does the author mean by "Actually, all of this takes place on the boundary"?

The crucial lemma, though, is:

Suppose that \(\displaystyle x_1, x_2, y_1, y_2 \in \partial \mathbb{H}^2\) and suppose that \(\displaystyle y_1, y_2\) tohether separate \(\displaystyle x_1\) from \(\displaystyle x_2\) in \(\displaystyle \partial \mathbb{H}^2\). Then \(\displaystyle f(y_1), f(y_2)\) together separate \(\displaystyle f(x_1)\) from \(\displaystyle f(x_2)\) in \(\displaystyle \partial \mathbb{H}^2\).

It is supposed to imply \(\displaystyle f\)'s continuity.

I think that this lemma may mean that however small "interval" \(\displaystyle (x_1, x_2)\) we'll take and \(\displaystyle y_1\) inside it, \(\displaystyle f(y_1)\) will end up inside a small "interval" \(\displaystyle (f(x_1), f(x_2))\).

Could you help me with my questions?

Thank you.