A theorem of Barbot (building on work of Ghys, Haefliger and others) says that Anosov flows on 3-manifolds are classified up to orbit equivalence by the data of a pair of transverse foliations of the plane and an action of the fundamental group of the 3-manifold. In recent work with T. Barthelmé, as well as C. Bonatti, S. Fenley and S. Frankel, we have been developing an abstract theory of Anosov-like group actions of bifoliated planes, applicable both to the study of flows and as an interesting class of foliation-preserving dynamical systems in its own right. This minicourse will explain some of this theory and the connections between flows and group actions in dimensions 1, 2 and 3. [-]

A theorem of Barbot (building on work of Ghys, Haefliger and others) says that Anosov flows on 3-manifolds are classified up to orbit equivalence by the data of a pair of transverse foliations of the plane and an action of the fundamental group of the 3-manifold. In recent work with T. Barthelmé, as well as C. Bonatti, S. Fenley and S. Frankel, we have been developing an abstract theory of Anosov-like group actions of bifoliated planes, applicable both to the study of flows and as an interesting class of foliation-preserving dynamical systems in its own right. This minicourse will explain some of this theory and the connections between flows and group actions in dimensions 1, 2 and 3.[-]

A theorem of Barbot (building on work of Ghys, Haefliger and others) says that Anosov flows on 3-manifolds are classified up to orbit equivalence by the data of a pair of transverse foliations of the plane and an action of the fundamental group of the 3-manifold. In recent work with T. Barthelmé, as well as C. Bonatti, S. Fenley and S. Frankel, we have been developing an abstract theory of Anosov-like group actions of bifoliated planes, applicable both to the study of flows and as an interesting class of foliation-preserving dynamical systems in its own right. This minicourse will explain some of this theory and the connections between flows and group actions in dimensions 1, 2 and 3. [-]

The study of the path-connectedness of the space of $C^{r}$ actions of $\mathbb{Z}^{2}$ on the interval [0,1] plays an important role in the classification of codimension 1 foliations on 3 manifolds. One way to deform actions is by conjugation. If an action can be brought arbitrarily close to the trivial one by conjugation, it is said to be quasi-reducible. In this talk, we will present and compare obstructions to quasi-reducibility in different regularity classes, and draw conclusions concerning the initial connectedness problem.[-]

The curve graph is a well-studied and useful tool to study 3-manifolds and mapping class groups of surfaces. The fine curve graph is a recent variant on which the full homeomorphism group of a surface acts in an interesting way. In this talk we discuss some recent results which highlight behaviour not encountered in the 'classical' curve graph. In particular, we will discuss the first entries in a dictionary between properties from surface dynamics and geometric properties of the action (and, while doing so, construct homeomorphisms acting parabolically). This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.[-]

It is well-known that a finitely generated group acts faithfully on the real line if and only if it is left-orderable. When this is the case, it is natural to study the possible representations of G into the group of homeomorphisms the real line. Several questions can be asked: how many dynamically distinct representations does G admit, and is there a 'nice' invariant that distinguishes such representations under (semi-)conjugacy? Which such representations can be conjugated into the group of diffeomorphisms (of a given regularity C^r)? Which such representations are rigid under perturbations, or when can two representations be deformed into one another by a continuous path? An object of great help in addressing such questions is the Deroin space of the group G, which is a compact space endowed with a flow which encodes all possible actions of G on the line (whose construction is based on work of Deroin-Kleptsyn-Navas-Parwani).
Which I will survey some results that address these questions for various groups, including finitely generated solvable groups and a class of groups which includes Thompson's group F. The talk is based on joint works with J. Brum, C. Rivas and M. Triestino. [-]

Given a finite-type surface, there are two important objects naturally associated to it. The first is the mapping class group and the second is the curve graph, which the mapping class group acts on via isometries. This action is well understood and has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. For instance, the elements acting loxodromically on the curve graph and precisely the pseudo-Anosov homeomorphisms. In this talk I'll discuss recent joint work with Carolyn Abbott and Nicholas Miller as well as a project with Sam Taylor regarding infinite-type mapping classes that act as loxodromic isometries on graphs associated to infinite-type surfaces. The aim of these projects is to work towards a Nielsen-Thurston type classification of mapping classes for infinite-type surfaces to understand which homeomorphisms are the generalizations of pseudo-Anosovs is in this setting.[-]

A (pseudo)-Anosov flow on a 3-manifold can be understood through its orbit space, a bifoliated plane with a natural action of the fundamental group of the manifold. In this minicourse, we will describe techniques to study the dynamics of these orbit space actions as a means to understand the topological theory and the classification of (pseudo)Anosov flows in dimension 3. This leads to a more general theory of 'Anosov-like' actions on bifoliated planes, which form a rich class of discrete dynamical systems including but not limited to the orbit space actions from flows.[-]