Sebastian Hensel - Homeomorphism groups of surfaces (in the fine curve graph)

Talk

Aim: Study surface homeos using GGT

We define the Fine curve graph. This is a graph with vertices for curves and edges for curves with $\le 1$ intersection. Why one intersection and not $0$?

Theorem (Bowden, H, Web)

The graph is Gromov hyperbolic. This is good since then we can get some properties on the group. Remember existence of Quasimorphisms.

Idea: Relate the action of $f$ on the space to the action on the fine curve graph. How is the study of the curve graph and the fine curve graph different?

Theorem (Bowden, H, Mann, Milton, Webb)

Let $f$ be a homei isotopic to the identity. The following are equivalent

1. $f$ acts loxodromically
2. There is a set of periodic points. If they are removed, then $f$ becomes a Pseudo-Anosov Homeomorphism.
3. The (homological ergodic) Rotation set of $f$ has nonempty interior.

Theorem (Bowden, Webb)

If in 2. $f$ is a pseudo-Anosov homeo, then the invariant foliation are identified with a quasiaxis in the fine curve graph.

Application

We can find specific elements with positive Stable commutator length. For that we use the Bestvina-Fujiwara machinery. Also there is a Tits alternative type of argument at play here.

Definition (Misiurewicz, Zimian)

We define the Rotation set as the average rate of escape on the fundamental region on the torus.

What next? (with Le Roux). Relate the shape of the rotation set to the geometry of the axis of $f$.

The rational points in the interior of the rotation set correspond with periodic points. The rotation set will always be compact and convex. It isn’t known however whether they can be the circle or not.