Abstract : It is well-known that a finitely generated group acts faithfully on the real line if and only if it is left-orderable. When this is the case, it is natural to study the possible representations of G into the group of homeomorphisms the real line. Several questions can be asked: how many dynamically distinct representations does G admit, and is there a 'nice' invariant that distinguishes such representations under (semi-)conjugacy? Which such representations can be conjugated into the group of diffeomorphisms (of a given regularity C^r)? Which such representations are rigid under perturbations, or when can two representations be deformed into one another by a continuous path? An object of great help in addressing such questions is the Deroin space of the group G, which is a compact space endowed with a flow which encodes all possible actions of G on the line (whose construction is based on work of Deroin-Kleptsyn-Navas-Parwani).
Which I will survey some results that address these questions for various groups, including finitely generated solvable groups and a class of groups which includes Thompson's group F. The talk is based on joint works with J. Brum, C. Rivas and M. Triestino.