Let \(I, J\) be connected open intervals such that \(I \cap J\) is a nonempty proper sub-interval of both \(I\) and\(J\). For instance, \(I = (0, 2)\) and \(J = (1, 3)\) form an example.

Let \(f\) (\(g\), resp.) be an orientation-preserving homeomorphism of the real line \(\mathbb{R}\) such that the set of points of \(\mathbb{R}\) which are not fixed by \(f\) (\(g\), resp.) is precisely \(I\) (\(J\), resp.).

Show that for large enough integer \(n\), the group generated by \(f^n, g^n\) is isomorphic to the group with the following presentation

\[ <a, b | [ab^{-1}, a^{-1}ba] = [ab^{-1}, a^{-2}ba^2] = id>. \]

The best solution was submitted by 김동률 (수리과학과 2015학번). Congratulations!

Here is his solution of problem 2019-12.