Mathematics

In this talk the goal is to present new results and questions
on an old idea initially due to R.Bowen and C.Series. In their paper,
back to 1979, they defined a single map associated to a group. This
particular map satisfies some nice dynamical properties, relating the
dynamics with the group.

I will discuss new constructions and results based on this idea. The
groups in question are, unfortunately, limited to surface groups, i.e.
fundamental group of a surface.

The associated maps belong to a class that is not so studied: expanding piecewise homeomorphisms of the circle.
The questions go in both directions: group and dynamics and there are
many challenges that I will try to explain.