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.

In this talk, I will introduce the notion of an end-periodic homeomorphism and its stretch factors motivated by the study of pseudo-Anosov homeomorphisms and their stretch factors. I will then discuss recent joint work with Paige Hillen and Chenxi Wu in which we characterize the set of end-periodic stretch factors. 

By a theorem of Agol and Guéritaud, every transitive pseudo-Anosov flow on a closed oriented 3-manifold can be combinatorially encoded by a finite veering triangulation. This encoding is not unique: different veering triangulations arise by drilling out different collections of periodic orbits of the flow.

In this talk, I will describe an algorithm that, given one veering triangulation, constructs another that encodes the same flow. I will also discuss the dynamical motivations for studying this operation and why understanding the resulting change in the combinatorics matters.

This is joint work with Henry Segerman.

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.

In this talk, we show that the map σ:Tg→Cg, which sends a compact hyperbolic surface X to a random simple closed geodesics on X, defines a proper embedding of Teichmüller space into the space of geodesic currents. In particular, we show that σ induces a compactification of Tg by projective measured laminations that agrees with Thurston’s compactification almost everywhere but differs from it at infinitely many points. This is joint work with Curt McMullen.

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.

Inspired by the well-studied case of hyperbolic surfaces, we can ask about the expected value of geometric properties of translation surfaces for large genus.

I will give an overview over a few results, most of them related to lengths of saddle connections and hence to Siegel-Veech constants. This is partly based on joint works with Howard Masur and Kasra Rafi.

The dynamics of horocycle flow in a hyperbolic surface goes back to Hedlund in the 1930's who showed for example that in a closed surface every horocycle is dense.  In the infinite-area case, a variety of phenomena can happen. For a Z^d cover of a compact surface S, there is an interesting connection between the behavior of orbit closures and the geometry of Thurston-type stretch laminations in S, as governed by the "stable norm" on the deck group. I will try to tell this story, which I hope is of interest to hyperbolic geometers as well as homogeneous dynamicists. This is joint with James Farre and Or Landesberg, and while some is work in progress, in many cases we have a complete classification of orbit closures and their Hausdorff dimensions.

Suppose that M is a complete, cusped, finite-volume oriented hyperbolic three-manifold.  Suppose that V is a transverse veering triangulation on M.  Then we produce a "Cannon-Thurston map" for V: a continuous equivariant surjection \Phi_V from the veering circle of V to the boundary of the universal cover of M.  If M is fibered, and the fiber is carried by V, then V is layered [Landry-Minsky-Taylor].  In this case \Phi_V "agrees" with the classical map [Bowditch, Mj].  If V is not layered then our map is new.

We also give an algorithm to draw approximations of \Phi_V.  In the layered case (where comparisons can be made), our algorithm gives ``better'' pictures for less ``work'' than Thurston's algorithm.

This is joint work with Jason Manning and Henry Segerman.