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

Abstract

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,Φ) 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,Φ) 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.