A realisation result for moduli spaces of group actions on the line

Abstract Given a finitely generated group , the possible actions of on the real line (without global fixed points), considered up to semi‐conjugacy, can be encoded by the space of orbits of a flow on a compact space naturally associated with and uniquely defined up to flow equivalence, that we call the Deroin space of . We show a realisation result: every expansive flow on a compact metrisable space of topological dimension 1, satisfying some mild additional assumptions, arises as the Deroin space of a finitely generated group. This is proven by identifying the Deroin space of an explicit family of groups acting on suspension flows of subshifts, which is a variant of a construction introduced by the second and fourth authors. This result provides a source of examples of finitely generated groups satisfying various new phenomena for actions on the line, related to their rigidity/flexibility properties and to the structure of (path‐)connected components of the space of actions.