What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$?

If the centers of the rotations coincide, then the rotations commute and generate some quotient of $\mathbb Z^2$ that can be understood by the number theoretic properties of the two rotation angles.

My guess would be that the group generated is always nonabelian free when the centers are distinct, but I don't have a proof. It seems difficult to understand in general because the group is indiscrete, so e.g. it might be hard to set up a ping pong argument to prove freeness.

NB that the relative position of the centers doesn't matter because you can always rotate and scale the picture. It's only a question of whether the centers are identical or not identical.

This came out of something I was thinking about in my research, but it's also related to this fascinating paper I just learned about

**Edit** Many pointed out that the group can never be free for a trivial reason: commutators of rotations give translations, so these groups will have plenty of commuting elements. In other words, these groups are metabelian. The next question would be whether the group generated is a free metabelian group on two generators. Or perhaps we ask whether the monoid generated is free (thereby forbidding commutators).

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