The broad theme of this minicourse is the relationship between the dynamics of a group acting on a space, the algebraic structure of the group, and some geometric structure responsible for the action; and how one of these can often determine the other two.  A paradigm example of this is the  "convergence group theorem" of Tukia, Gabai, and Casson--Jungries, which says that a convergence group (a purely dynamical condition) acting on the circle is a Fuchsian group, i.e isomorphic to the fundamental group of a hyperbolic surface or orbifold; and the action on the circle is the action on the boundary of the surface or orbifold's universal cover.  

In a similar spirit, Thurston defined an "extended convergence group" for groups acting on the line, and showed using ideas of Barbot and Fenley that these are always fundamental groups of 3-manifolds, and the action on the line comes from an Anosov flow on the 3-manifold.   However, not all 3-manifolds with Anosov flows give rise to extended convergence groups; instead giving rise to a more general class called "Anosov-like actions on bi-foliated planes"

In the course, we'll study Anosov-like actions from the axioms, and see how some simple dynamical assumptions can lead to a rich structure theory.  This is a perspective that Barthelmé, Bonatti, Fenley, Frankel and I (in various combinations and in still ongoing work) have recently successfully used towards the study and classification of Anosov flows on 3-manifolds, building on work of Barbot, Fenley, and others.