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Anosov and pseudo-Anosov flows are a type of hyperbolic flows that have been studied since the 1960s from a dynamical point of view, and at least the 80s from a topological point of view. They have attracted renewed interest, in particular in dimension 3, in the recent past due to the many connections between dynamics, geometrical and topological aspects of the flows and their supporting manifolds.

A very powerful tool for the topological study of pseudo-Anosov flows, introduced in the 90s by Barbot and Fenley, is the induced action of the fundamental group on the \emph{orbit space} of the flow.

With Kathryn Mann, and different other collaborators, including Christian Bonatti, Sergio Fenley and Steven Frankel, we introduced an axiomatization of that induced action.

From this point of view, the study of flows consists in using, and building up, the dictionary between dynamical/ topological properties of the flow and that of the induced action, as well as finding new algebraic/topological/dynamical properties of such class of actions.

In this talk, I will discuss some of the recent progress we obtained in this manner.

While I will recall the notion of pseudo-Anosov flows to motivate this study, a large part of the talk will deal with much more basic objects and should be accessible to anyone who knows what a group is and is willing to draw lines on a plane.