Workshop

The fundamental theorem of Thurston states that any homeomorphism of a
surface of finite type can be isotoped so that some multi-curve is
invariant and so that for every complementary component the first
return map is either periodic or pseudo-Anosov. Homeomorphisms of
infinite type surfaces are much more complicated. In this work we
focus on the class of tempered homeomorphisms -- these are the ones
that do not have any mixing behavior. We show that up to isotopy
there is an invariant geodesic lamination so that the first return maps
display one of three qualitatively different behaviors. This work is in progress and it is
joint with Federica Fanoni and Jing Tao.

How do you measure the difference between two hyperbolic surfaces? In the 80s, Thurston proposed a new version of Teichmüller theory that says to look at the smallest Lipschitz constant of maps between them. He proved that maps with the best possible constant exist, and while they are not unique, minimizers are always rigid along a geodesic lamination. In this talk, I’ll describe work with Jing Tao in which we coarsely characterize what must (and what can) happen on the rest of the surface.

Given a fixed finite depth foliation F of an atoroidal manifold M, Gabai and Mosher described how to build a pseudo-Anosov flow on M that is almost transverse to F. The construction is somewhat flexible, and hence some foliations are transverse to multiple inequivalent pseudo-Anosov flows. However, Mosher’s Transverse Finiteness Conjecture (1996) asserts that a given finite depth F can be almost transverse to only finitely many pseudo-Anosov flows. In joint work with Michael Landry, we prove this conjecture. We also prove a related rigidity result: if F is almost transverse to some pseudo-Anosov flow without perfect fits, then it is in fact the unique pseudo-Anosov flow transverse to F. In our proofs, we organize possible flows by reconstructing their universal circles inductively through the depths of F, using Cannon—Thurston maps to control the possible choices that arise in the reconstruction.

The fine curve graph is a variant of the curve graph for the homeomorphism group: a Gromov hyperbolic graph on which the homeomorphism group acts. In this talk we present joint work with Frédéric Le Roux linking the shape of the rotation set of a torus homeomorphism (a classical, dynamical conjugacy invariant) to the geometry of its action on the fine curve graph. As a consequence, one can construct homeomorphisms with positive scl close to the identity, and obtain Tits alternatives for (certain) subgroups of the homeomorphism group of the torus.