Biringer - Growth of Betti numbers in higher rank locally symmetric spaces

Let X be a higher rank irreducible symmetric space of non-compact type. We will show that the growth of Betti numbers in any sequence of distinct, compact X-manifolds with injectivity radius bounded below is controlled by the L2 Betti numbers of X.  The key technique is a probabilistic version of Gromov-Hausdorff convergence, adapted from Benjamini-Schramm convergence in graph theory.

Joint work with Abert, Bergeron, Gelander, Nikolov, Raimbault, Samet.

- The Weil-Petersson metric and the geometry of hyperbolic 3-manifolds

Many coarse relationships support the existence of a deep connection between the synthetic geometry of the Weil-Petersson metric and the geometry of hyperbolic 3-manifolds.  In this talk, I'll elaborate on some new, finer examples of such connections, as well as some contrary evidence to a fundamental link. This talk will present joint work with Yair Minsky and Juan Souto.

- Dynamics on character varieties

If $\Gamma$ is a finitely presented group and $G$ is a Lie group, it is natural to study the dynamics of the action of $Out(\Gamma)$ on the $G$-character variety of $\Gamma$. In this talk, we will briefly survey previous work and then focus on recent work of Canary, Lee, Magid, Minsky and Storm in the setting where $\Gamma$ is a 3-manifold group and $G=PSL(2,C)$.

Chatterji - Distortion and bounded cohomology for Lie groups

I will explain how the distortion of central subgroups in Lie groups is related to the Borel cohomology in degree 2, and illustrate a few results on examples. This is joint work with Mislin, Pittet and Saloff-Coste, as well as work in progress with Cornulier, Mislin and Pittet.

- The geometry of right-angled Artin subgroups of the mapping class group

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasi-isometric embedding for both of the standard metrics.

- Length functions for Hitchin representations

Let S be a closed oriented surface of negative Euler characteristic. We consider the space Rep_n(S) of conjugacy classes of homomorphisms from the fundamental group of S to PSL_n(R), with n > 2. Using Higgs bundle techniques, N. Hitchin described the number of connected components of Rep_n(S). In particular, he gave a parametrization of one connected component, called the Hitchin space, which contains a copy of the Teichmüller space of S. Given a closed curve c on S and a representation r in the Hitchin space of S, we can consider the eigenvalues of r(c). We first show how to extend these eigenvalue functions to length functions on the space of measured geodesic currents on S, or more generally on the space of Hölder geodesic currents. Then we introduce cataclysm deformations for Hitchin representations and study the effect of these deformations on the length functions of a Hitchin representation.

This work is based on Labourie's dynamical characterization of Hitchin representations.

- Complex projective surfaces bounding 3-manifolds

A number of properties of the space of complex projective structures on surfaces bounding a 3-manifold will be derived from Poincare duality.  A simple formula for the Goldman complex symplectic structure implies Kleinian and quasi-Fuchsian reciprocity and the lagrangian nature of various sections, such as Bers slices.

- Exponential decay in the mapping class group

We show that a random walk on the mapping class group gives a non-pseudo-Anosov element with a probability which decays exponentially in the length of the random walk. More generally, we show that the probability that a random walk gives an element with bounded translation distance on the curve complex decays exponentially.

Tao -
Geodesics in Teichmuller Space Equipped with Thurston's Lipschitz Metric

Thurston defined an asymmetric metric on Teichmuller space using Lipschitz maps and proved that distances can be computed using hyperbolic length ratios between curves. This contrasts with the Teichmuller metric on Teichmuller space, defined using quasi-conformal maps, which by Kerckhoff's formula can be computed using extremal length ratios between curves. In joint work with Anna Lenzhen and Kasra Rafi, we study geodesics in the Lipschitz metric. We show that if the endpoints of a Lipschitz geodesic have bounded combinatorics, then it fellow travels the unique Teichmuller geodesic with the same endpoints. For arbitrary Lipschitz geodesics, we show that their projection to the curve complex are unparametrized quasi-geodesics. The proof in the latter is inspired by the recent work of Bestvina-Feighn on the hyperbolicity of the free factor complex.

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