  Given a finitely generated group $G$, the possible actions of $G$ 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 $(Y, \Phi)$
naturally associated with $G$ and uniquely defined up to flow equivalence, that
we call the \emph{Deroin space} of $G$. We show a realisation result: every
expansive flow $(Y, \Phi)$ 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.
