How to describe the all possible actions on the line of a given group? The quotient obtained by considering all actions without global fixed points, up to semi-conjugacy can be bad in topological terms.
For finitely generated groups one can consider an intermediate quotient given by the collection of harmonic actions of the group. This turns out to be a compact space endowed with a translation flow, so that two actions are semi-conjugate if and only if they are in the same orbit of the flow.

We are able to understand the space of harmonic actions for solvable groups and many locally moving groups (including Thompson's F and generalizations): the actions of these groups which are not the obvious ones, are all obtained from actions on planar real trees fixing a point at infinity. This talk is based on a joint project with J Brum, N Matte Bon and C Rivas.