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Given a finitely generated group $G$, the possible actions of $G$ on the realline (without global fixed points), considered up to semi-conjugacy, can beencoded by the space of orbits of a flow on a compact space $(Y, \Phi)$naturally associated with $G$ and uniquely defined up to flow equivalence, thatwe call the \emph{Deroin space} of $G$. We show a realisation result: everyexpansive flow $(Y, \Phi)$ on a compact metrisable space of topologicaldimension 1, satisfying some mild additional assumptions, arises as the Deroinspace of a finitely generated group. This is proven by identifying the Deroinspace of an explicit family of groups acting on suspension flows of subshifts,which is a variant of a construction introduced by the second and fourthauthors. This result provides a source of examples of finitely generated groupssatisfying various new phenomena for actions on the line, related to theirrigidity/flexibility properties and to the structure of (path-)connectedcomponents of the space of actions.

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