Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim\_{n \to \infty} \tilde{f}^n(0)/n) + \mathbb{Z}$, where $\tilde{f}: \mathbb{R} \to \mathbb{R}$ is any lift of $f$. Intuitively, it measures the rate of circulation around the circle.


Now $\rho: Homeo\_+(S^1) \to \mathbb{R}/\mathbb{Z}$ is not a homomorphism, but its restriction to any amenable subgroup of $Homeo\_+(S^1)$ is. Thus if $G$ is amenable, and $\phi: G \to Homeo\_+(S^1)$ is an action, then the composition $\rho \circ \phi: G \to \mathbb{R}/\mathbb{Z}$ is a homomorphism.


I am wondering, what homomorphisms $G \to \mathbb{R}/\mathbb{Z}$ arise as $\rho \circ \phi$ if $\phi$ is required to be 1-1?


(The requirement that $\phi$ be 1-1 is important, since otherwise we can realize any $\psi: G \to \mathbb{R}/\mathbb{Z}$ simply by making $G$ act on the circle via $\psi$.)


This question might be hard, since I don't even know which amenable groups act faithfully on the circle. But I'd even be curious about the case where $G$ is a finitely-generated, torsion-free nilpotent group. These guys do act on the circle faithfully, but the standard construction gives you something trivial in terms of rotation number.


I doubt it helps, but by Ghys and Matsumoto, for an amenable group $G$, two actions $\phi\_1, \phi\_2: G \to Homeo\_+(S^1)$ are semi-conjugate if and only if $\rho(\phi\_1(g)) = \rho(\phi\_2(g))$ for every $g \in G$ (Matsumoto, "Numerical invariants for semiconjugacy of homeomorphisms of the circle").



Rotation numbers for amenable group actions on the circle
The answer to my question, if $G$ is assumed to be a finitely generated torsion-free nilpotent group, is that *any* homomorphism $\psi: G \to \mathbb{R}/\mathbb{Z}$ can be realized as $\rho \circ \phi$ for $\phi: G \to Homeo\_+(S^1)$ 1-1. I'm sure this was already known, and the paper referenced by Dan Sălăjan definitely contains the right way to think about it.


Suppose we have $\psi: G \to \mathbb{R}/\mathbb{Z}$. Take the orbit of some $x \in S^1$ under $im(\psi)$, and blow up the points in this orbit so that they are intervals. We could call the circle with blown-up intervals $\tilde{S^1}$. Obviously, we can define an action of $im(\psi)$ on $\tilde{S^1}$, for instance by sending one interval to another affinely.


Now $G$ is a f.g. torsion-free nilpotent group, so it acts faithfully on the interval $[0, 1]$, and thus so does $\ker(\psi)$. If we take a copy of this action of $\ker(\psi)$ on each blown-up interval, together with the action of $im(\psi)$ that sends one interval to another, then altogether we have an action of the wreath product $\ker(\psi) \wr im(\psi)$ on $\tilde{S^1}$.


By the Kaloujnine-Krasner Theorem, $\ker(\psi) \wr im(\psi)$ contains a copy of every extension of $\ker(\psi)$ by $im(\psi)$, including $G$ itself. Indeed, under this embedding $\Phi\colon G \hookrightarrow \ker(\psi) \wr im(\psi)$, the projection of $\Phi(g)$ onto the $im(\psi)$ factor is precisely $\psi(g)$. (For more information on the Kaloujnine-Krasner Theorem, see e.g. Lectures on Finitely Generated Solvable Groups, in SpringerBriefs in Mathematics.) So this allows us to define our embedding $\phi\colon G \to Homeo\_+(S^1)$ having the desired property.


