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").