Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.

In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.

In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.

The talks are based on joint work of the speakers with J. Brum and C. Rivas.

Given a group G, we are interested in describing the possible actions of G on the real line, that is its representations into the group of homeomorphisms of the real line. In this series of two talks, we present a setup that allows to give a satisfactory picture for a class of groups, which arise as sufficiently rich subgroups of the group of homeomorphisms of the line, called locally moving groups. This class contains various finitely generated groups acting on the line: a famous example is Thompson’s group F.

In this first talk, we will focus on action by C^1-diffeomorphisms. In particular we will see that if G is a locally moving group of homeomorphisms on the line, then every faithful minimal action of G on the line is topologically conjugate to its natural defining action.

In the second talk, we will focus on actions that are only by homeomorphisms, that turn out to be much more flexible: for example we will explain that Thompson’s group F admits a continuum of exotic minimal faithful C^0 actions on the line. Nevertheless we will see that such actions have a very special structure: this will lead us to introduce the notion of R-focal action, a class of actions on the line that can be suitably encoded by certain actions on planar real trees. As an application we will obtain a local rigidity result: for a vast class of locally moving groups, all small perturbations of the natural defining actions are semi-conjugate to it.

The talks are based on joint work of the speakers with J. Brum and C. Rivas.

Two topological dynamical systems \((X, T, G)\) and \((Y,S,\Gamma)\) are said to be (topologically) orbit equivalent if there exists a homeomorphism \(h:X\to Y\) sending \(T\)-orbits onto \(S\)-orbits. A general question in the framework of orbit equivalence is how the dynamical properties of a given system are related to its orbit equivalence class. In this talk we will discuss about what are the natural restrictions that arise from being orbit equivalent, in the case of minimal shift actions on the Cantor set given by countable amenable groups. We will also discuss about the problem of realization of Choquet simplices as sets of invariant measures of topological dynamical systems and see how this is connected to the problem of classification up to orbit equivalence.

We prove that for each finite index subgroup \(H\) of the mapping class group of a closed hyperbolic surface, and for each real number \(r>0\) there does not exist a faithful \(C^{1+r}\)--action of \(H\) on a circle. (Joint with Thomas Koberda and Cristobal Rivas).

Let \(\Gamma\) be the fundamental group of a closed (real) hyperbolic \(n\)-manifold \(M.\) We study the second variation of the Hausdorff dimension of the limit set of convex co-compact morphisms acting on the complex-hyperbolic space \(\rho:\Gamma\to Isom(\mathbb H^n_\mathbb C)\), obtained by deforming a discrete and faithful representation of \(\Gamma\) that preserves a totally geodesic (and totally real) copy of the real-hyperbolic space \(\mathbb H^n_\mathbb R\subset\mathbb H^n_\mathbb C\). This computation is based on the study of the space of convex projective structures on \(M\) and a natural metric on it induced by the Pressure form. This is joint work with M. Bridgeman, B. Pozzetti and A. Wienhard.

A hyperbolic group acts on its Gromov boundary by homeomorphisms. In recent joint work with Jason Manning, we show these actions are topologically stable whenever the boundary is a sphere: any small perturbation of the action is semi-conjugate to the original action. This is also true for free groups, with cantor set boundary, and in ongoing work we are investigating the general case. In my talk, I will explain some of the strategy of the proof and motivation for the problem.

Let G be a group acting on a finite dimensional CAT(0) cube complex X. By studying equivariant maps from the Furstenberg-Poisson boundary to the Roller boundary, we deduce a variety of phenomena concerning the push-forward of the random walk from G to an orbit in X. Under mild and natural assumptions, we deduce positivity of the drift, sublinear tracking, and a central limit theorem. Along the way we prove that regular elements are plentiful and establish a homeomorphism between the boundary of the contact graph of X with a special subset of the Roller boundary called the regular points. This is joint work with Jean Lécureux and Frédéric Mathéus.