Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime.
In the book, by Pierre de la Harpe, Topics in Geometric Group Theory section $\textrm{II.B.40}$, as a research problem, it asks to find an appropriate "ping-pong" action to show that the group, under function composition, $G=\langle f,g \rangle$ is a free group of rank two.
Is there such a proof? That is, is there a proof where the key insight is having that group act in such a way to apply the ping-pong lemma(table-tennis lemma)? I have not been able to find such a proof either by working on it, or in the literature.
Maybe we don't have such a proof but do we have a proof that $G$ contains a free subgroup of rank two, akin to proofs for torsion-free hyperbolic group, or the Tits alternative. I am not sure how obvious it is that $G$ is hyperbolic, or linear. I am guessing it is not obvious that it is linear since I would suspect a ping-pong proof would come out of that pretty quickly.
Note that there are proofs of this theorem, but as far as I know, they do not use the ping-pong lemma.
The only proofs of the result(and more general things) I know of are in :
Free groups from fields by Stephen D. Cohen and A.M.W. Glass
The group generated by $x \mapsto x+1$ and $x \mapsto x^p$ is free. by Samuel White
Arithmetic permutations by S.A. Adeleke and A.M.W. Glass