Spring 2022

Abstract: A group is coherent if every finitely generated subgroup is finitely presented. A group is incoherent otherwise. I’ll discuss some old (and recent!) results in both directions, and explain the deep connection to the geometry of subgroups.

Abstract: Rigidity theorems prove that a group’s geometry determines its algebra, typically up to virtual isomorphism. We study graphically discrete groups, which impose a discreteness criterion on the automorphism group of any graph the group acts on geometrically. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds. We will present new examples and discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherrd, and Daniel Woodhouse.

Abstract: (Joint work with Ashani Dasgupta)
The boundary of any one-ended hyperbolic group is always locally connected.  This deep result, combining work of Bestvina-Mess, Levitt, Bowditch, and Swarup, was one of the first major results about hyperbolic groups, established in the 1990s.  This theorem led to advances in JSJ decompositions, boundary classification problems, and the analytic study of boundaries.  Shortly afterward, Bowditch showed how to associate a natural boundary to any relatively hyperbolic group pair.  Bowditch conjectured that this boundary is always locally connected, and proved this fact in the presence of certain awkward hypotheses: the group in question must be finitely presented, hyperbolic relative to 1- or 2-ended groups, and have no infinite torsion subgroups.


In fact, the boundary of a one-ended relatively hyperbolic group pair is always locally connected, with no restrictions.  This result applies in a general setting, in which the groups in question need not be finitely generated or even countable.

Abstract: In this talk, we will relate homological filling functions with coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result.

Abstract: The Nielsen-Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form.  I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour.  This forms a significant barrier to translating specific arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

Abstract: A spectre is haunting Geometric Group Theory — the spectre of a generalized Leighton’s Graph Covering Theorem. The original theorem states that any two graphs with common universal cover have a common finite cover. Haglund conjectured that this should generalize to all compact special cube complexes. I will talk about recent progress on this, my own contributions alongside others. I will discuss the implications for quasi-isometric rigidity, and for hyperbolic groups in particular. I will give some conjectures and explain why they should be true and very loosely how (other people) will likely one day prove them.

Abstract: It is a consequence of a well-known result of Ahlfors and Bers that the character associated to a convex co-compact hyperbolic 3-manifold is determined by its peripheral data. In this talk we will show how this map extends to a birational isomorphism of the corresponding character varieties, so in particular it is generically a 1-to-1 map. Analogous results were proven by Dunfield in the single cusp case, and by Klaff and Tillmann for finite volume hyperbolic 3-manifolds. This is joint work with Ian Agol.

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