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.