In this talk, I will discuss the moduli space of actions of a finitely generated group G on the real line. I will first explain what we mean by this space, and then describe its structure in the cases of solvable and Thompson-like groups. Finally, if time permits, I will present some open problems concerning the structure of these spaces for amenable groups.

This is a joint work with Nicolás Matte Bon, Michele Triestino and Cristóbal Rivas.