Fall 2023

Abstract:  I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. Milnor showed that any such 3-manifold admits a unique decomposition as a connected sum of prime manifolds. We exploit this decomposition on the level of moduli spaces, and consequently we prove that when M has non-empty boundary, B Diff (M rel \partial M) has the homotopy type of a finite CW-complex (as conjectured by Kontsevich). We also compute the rational cohomology of B Diff ((S^1 x S^2) # (S^1 x S^2)).

Abstract:  An old problem in dynamical systems is to try to classify Anosov flows up to orbit-equivalence. This question is particularly interesting in dimension 3 where we both have lots of examples and a rich, but still poorly understood, relationships between the dynamics of the flow and the topology of the manifold. By a result of T. Barbot, classifying Anosov flows (or more general pseudo-Anosov flows) in dimension 3 up to orbit equivalence restricts to classifying, up to conjugacy, certain actions of \pi_1(M) on the orbit space, a topological plane with two transverse foliations.  
In this talk, I will recall the above and discuss a new complete invariant for transitive (pseudo)-Anosov flows which often reduces to just knowing which conjugacy classes in \pi_1(M) are represented by periodic orbits of the flow. 
This is all joint work with Kathryn Mann and Steven Frankel.

Abstract:  How can one detect whether a particular element in a group belongs to the k-th term in the lower central series? And, given two such elements, how can one distinguish them?
We define invariants of words in arbitrary groups, measuring how letters in a word are linked, and distinguishing elements up to high-order commutators. These are the letter-linking invariants. On free groups, linking invariants coincide with coefficients in the Magnus expansion, but they are defined on all groups and over any PID, effectively giving analogous expansions in general. They respect products in the group and are complete, so serve as the coefficients of a universal multiplicative finite-type invariant, depending functorially on the group.

Furthermore, letter-linking invariants carry both combinatorial meaning and a geometric interpretation relative to a topological model. As an application, we define a Johnson style filtration and a Johnson homomorphism on the automorphisms of any group.

Abstract:  A trace on a group is a positive-definite conjugation-invariant function on it. These functions play an important role in harmonic analysis of discrete groups, and their study has found many exciting connections to rigidity, stability, and dynamics in the past couple of decades. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will then see the different behaviours of these spaces for free groups vs. higher-rank lattices. Finally, some open questions about free products and surface groups will be presented. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

Abstract:  We prove that for n≥2, a non-uniform lattice in PU(n,1) does not admit a relatively geometric action on a CAT(0) cube complex, in the sense of Einstein and Groves. As a consequence, we classify which lattices in semisimple Lie groups can admit relatively geometric actions on CAT(0) cube complexes.  This is joint work with Daniel Groves and Kejia Zhu.  

Abstract:  (Based on a paper written jointly with Kim Ruane.)

The goal of this talk is to give an elementary exposition of some results due to Bestvina–Mess on the local connectivity of the boundary of a one-ended hyperbolic group.  Along the way we will show that one-ended hyperbolic groups are semistable at infinity and their boundaries are linearly connected.  Geoghegan first observed that local connectivity implies semistability using some deep results of shape theory.  Bonk–Kleiner originally proved linear connectedness of the boundary using subtle analytic methods.  We show how all three results follow directly from elementary methods inspired by the proofs of Bestvina–Mess.

Abstract:  It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups.  This is joint work with Jason Behrstock and Jacob Russell. 

Abstract:  Regular trees of graphs are inverse limits of particularly simple inverse systems of finite graphs.  They form a 1-dimensional subclass of the Markov compacta: a class of finitely describable inverse limits of simplicial complexes, which includes all boundaries of hyperbolic groups.  I will discuss upcoming joint work with Jacek Swiatkowski in which we use Bowditch’s canonical JSJ decomposition to characterize the 1-ended hyperbolic groups whose boundaries are (regular) trees of graphs.