In 1999, E. Klarreich found a very intriguing correspondence between the Gromov boundary of the curve graph for closed surfaces (a very GGT object) with the space of ending laminations on the surface (a very geometric object). Since then, Hamendstädt, Schleimer and Pho-On have thought about various proofs for this result, and generalizations to the arc graph / the arc-and-curve graph for finite-type surfaces. The grand arc graph is a type of arc graph associated with certain infinite-type surfaces, which is also an infinite-diameter hyperbolic graph. In this talk, we shall talk about a couple of ways to define “laminations that should correspond to points on the Gromov boundary of the grand arc graph”. This work is joint with Carolyn Abbott and Assaf Bar-Natan.

Part of the Zimmer program aims to classify smooth actions of higher-rank lattices on manifolds. In a series of papers, Brown, Fisher and Hurtado showed that higher-rank lattices act ‘trivially’ on low dimensional manifolds. For the lowest dimension allowing nontrivial and chaotic actions, one can expect rigidity. Lee proves such a rigidity result for volume preserving Anosov actions of lattices in Sl(n,R) on n-dimensional closed manifolds: they are (up to finite error) conjugate to automorphisms of the torus. The proof relies on the existence of many Anosov elements for the model action. For the standard toral actions associated with lattices in SO(n,n+1), however, every matrix has eigenvalue 1. In particular, it has no Anosov elements. Nevertheless, we show in this talk that partially hyperbolic volume preserving actions of lattices in SO(n,n+1) on (2n+1)-dimensional manifolds are (up to finite error) conjugate to the action by automorphisms of the torus. This is joint work with Homin Lee.

The topology and geometry of foliations on 3-manifolds often inform the dynamics of transverse flows. In this vein, we address the following question: What properties of a foliation force all transverse flows to have closed orbits? We fully resolve this question for C2 finite depth foliations on hyperbolic 3-manifolds. These foliations generalize finite-type fibrations by allowing for a hierarchy of spiraling behavior. For the foliations which don’t force closed orbits, we construct aperiodic transverse flows. This is joint work with Audrey Rosevear.

TBD