14.8 (1999)

Let $G$ denote the group of germs at $+\infty$ of orientation-preserving homeomorphisms of the real line $\mathbb{R}$. Let $\alpha \in G$ be the germ of $x \mapsto x + 1$. What are the germs $\beta \in G$ for which the subgroup $\langle \alpha, \beta \rangle$ of $G$ generated by $\alpha$ and $\beta$ is free of rank 2?

If $\beta$ is the germ of $x \mapsto x^k$ for an odd integer $k \geqslant 3$, it is known that $\langle \alpha, \beta \rangle$ is free of rank 2. The proofs of this rely on Galois theory (for $k$ an odd prime: S. White, J. Algebra, 118 (1988), 408–422; for any odd $k \geqslant 3$: S. A. Adeleke, A. M. W. Glass, L. Morley, J. London Math. Soc., 43 (1991), 255–268, and for any odd $k \neq \pm 1$ and any even $k > 0$ in several papers by S. D. Cohen and A. M. W. Glass).