Agol’s veering triangulation for 3-manifolds that fiber over the circle can be obtained very explicitly, via a construction of Gueritaud, from the stable and unstable foliations of the monodromy. We study the way in which these triangulations interact with the arc complexes of the surface and its subsurfaces. This allows us to examine the “profile” of subsurface projections associated to each fiber in a fibred face of the Thurston norm ball, finding explicit relations between these projections and the simplicial structure of the triangulations and obtaining some bounds that do not depend on the complexity of the fibers. Joint work with Sam Taylor

Relative train tracks, introduced by Bestvina-Handel, are convenient representatives of free group outer automorphisms. They serve as analogues of Thurston’s normal form for surface mapping classes. I will describe joint work with Michael Handel using particularly nice relative train tracks called completely split relative train tracks, or CTs for short, to compute some invariants of free group automorphisms.

The thick part of moduli space is compact and its  rich complex geometry if is often ignored while studying the coarse geometry of Teichmüller space. We examine the systole function on moduli space which is known to be a morse function and we will show that is has many local maxima. We will also review some older results and discuss some open questions.  

We will discuss a new algorithm for computing geodesics in the curve complex. This uses a refinement of the ideas of Masur-Minsky, Leasure, Shackleton, Watanabe and Webb to ensure such geodesics are found in polynomial time (in terms of their length). One corollary of this is a new (polynomial-time) algorithm to determine the Nielsen--Thurston type of a mapping class. This is joint work with Richard Webb.

Using the framework of hierarchical hyperbolicity, we introduce a new compactification of the mapping class group which possesses the robust dynamical features of Thurston's compactification of Teichmuller space but is better suited for studying the asymptotics of subgroups.  As an application, we define a notion of geometrical finiteness for subgroups of mapping class groups and prove that several well-studied classes of subgroups are geometrically finite.  This is joint work with Mark Hagen and Alessandro Sisto.

In this talk I shall overview various fixed point properties and their occurence in the classes of hyperbolic groups, and of higher rank lattices. I shall then present the existing evidence, in the setting of random groups, in support of a connection between the geometry of the boundary of a hyperbolic group and the fixed point properties that the group has. This is joint work with John Mackay.