Évènement

To every topological surface one can associate a group called the mapping class group (sometimes called the modular group), consisting of homeomorphisms of the surface up to isotopy. In the case of the torus, this group is SL(2,Z), whose elements fall into three dynamical types detected by the trace: elliptic (|tr A| < 2), parabolic (|tr A|=2), and hyperbolic (|tr A| > 2).

Remarkably, Thurston showed that an analogous classification exists for maps of finite-type surfaces: after cutting along a canonical multicurve, each component carries a map that is either periodic or pseudo-Anosov, and the map is reducible precisely when the reducing multicurve is nonempty. This mirrors the role of Jordan blocks in linear algebra.

In this talk, I will first present these classical examples—from the linear case of 2×2 matrices to the nonlinear case of finite-type surfaces—before discussing joint work with Mladen Bestvina and Federica Fanoni, where we investigate how this classification extends (or fails) for more general surfaces.