In the appendix here, we make the claim that the semigroup

is free on the given generators.

Here, I will give a proof. It is a simpler version of the ping-pong argument commonly used to prove that a group is free on a given set of generators.

First a simple observation. it suffices to prove that the set

generates a free semigroup.

Now, consider the action of our semigroup on the interval by fractional linear transformations:

In particular, note that

Now suppose that

Apply both sides to some . The LHS lies in and the RHS lies in . Therefore and the rest is an easy induction.