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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., Comment: 32 pages; v2: minor changes

We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an action on the line by affine transformations. We show that this is always the case for actions by $C^1$ diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms, for which this result is no longer true (as shown by Plante), we show that a semi-conjugacy to an affine action still exists in a local sense, at the level of germs near the endpoints. Finally for a vast class of solvable groups, including all solvable linear groups, we show that the family of affine actions on the line is robust, in the sense that any action by homeomorphisms on the line which is sufficiently close to an affine action must be semi-conjugate to an affine action. This robustness fails for general solvable groups, as illustrated by a counterexample., Comment: 30 pages

The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions of these groups. More precisely, suppose that $G$ is a subgroup of the homeomorphism group of the circle $\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree $\mathsf{Aut}(\mathbb{T})$. Under natural assumptions on the stabilizers of the action of $G$ on $\mathbb{S}^1$ or $\partial \mathbb{T}$, we use the dynamics of this action to show that every faithful action of $G$ on a set that is at least $3$-transitive must be conjugate to the action of $G$ on one of its orbits in $\mathbb{S}^1$ or $\partial \mathbb{T}$., Comment: 26 pages. v1-> v2: addition of an appendix and some minor corrections, v2->v3: Minor corrections and abstract rewritten. Final version to appear in the journal of the London Math. Soc

We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other groups of homeomorphisms, and implies that arbitrary homomorphisms between the full groups of a vast class of pseudogroups must extend to continuous morphisms between pseudogroups (in particular giving rise to equivariant maps at the level of spaces). As applications, we obtain explicit obstructions to the existence of embeddings between full groups in terms of invariants of the underlying pseudogroups (the geometry of their orbital graphs, the complexity function, dynamical homology), and provide a complete descriptions of all homomorphisms within various families of groups including the Higman-Thompson groups (more generally full groups of one sided shifts of finite type), full groups of minimal $\mathbb{Z}$-actions on the Cantor set, and a class of groups of interval exchanges. We next consider a combinatorial rigidity property of groups, which formalises the inability of a group to act on any set with Schreier graphs growing uniformly subexponentially, or more generally not faster than a given function $f(n)$. For the exponential function this is a well-known consequence of property $(T)$. We use full groups to provide a source of examples of groups which satisfy this property but satisfy a strong negation of property $(T)$. A key tool used in the proofs is the study of the dynamics of the conjugation action of groups on their space of subgroups, endowed with the Chabauty topology. In particular we classify the confined and the uniformly recurrent subgroups of full groups., Comment: 70 pages. Completely rewritten version, with a different formulation of the main result, and several new results. The title has changed. Comments are welcome!

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group $F$ is non-amenable, then Thompson's group $T$ must be $C^\ast$-simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups $F$, $T$ and $V$, for which we also deduce rigidity results for their minimal actions on compact spaces., Comment: 45 pages. Minor changes