Fall 2025

Abstract: I will discuss joint work with Matthew Cordes and Annette Karrer on Morse hierarchies. I will state the main theorem and give motivation for why we proved it, but will spend most of the time on finding Morse hierarchies for specific groups or classes of groups.

Abstract: We construct two convex cocompact groups of isometries of real hyperbolic spaces with limit set a Pontryagin sphere. Both examples arise from right-angled Coxeter groups and admit special subgroups with limit set a Menger curve. One example is in H^4 and the other is 2-generated in H^6, which makes these examples optimal with respect to dimension and rank respectively. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.

Abstract: If a group contains a strongly contracting element, then it is acylindrically hyperbolic. Moreover, one can use the Projection Complex of Bestvina, Bromberg and Fujiwara to construct an action on a hyperbolic space where said element acts loxodromically. However, the action depends on the chosen element and other strongly contracting elements are not necessarily loxodromic. It raises the questions whether there exists a single action on a hyperbolic space where all strongly contracting elements act loxodromically. In this talk, we answer the above question positively by introducing the contraction space construction.

Abstract:  A cobordism W between compact manifolds M and M’ is an h-cobordism if the inclusions of M and M’ into W are both homotopy equivalences. This kind of cobordism plays an important role in the classification of high-dimensional manifolds, as h-cobordant manifolds are often diffeomorphic.

With this in mind, given two h-cobordant manifolds M and M’, how different can their diffeomorphism groups Diff(M) and Diff(M’) be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees. Contrasting this fact, I will present examples of h-cobordant manifolds with different mapping class groups. In doing so, I will review the classical theory of h-cobordisms and introduce several moduli spaces of manifolds that help in answering this question.

Abstract: In this talk, I introduce a random model for an n-fold branched cover of a finite 2-complex X with mild hypothesis, and investigate its structural and probabilistic properties. In particular, we show that as n goes to infinity, a random branched cover asymptotically almost surely is homotopy equivalent to a 2-complex satisfying geometric small cancellation. The research presented in this talk is joint work with Jean-Francois Lafont and Rachel Skipper.

Abstract: I will discuss recent work with Minsky and Sisto, in which we prove that mapping class groups of finite-type surfaces—and more generally, colorable hierarchically hyperbolic groups (HHGs)—are asymptotically CAT(0). This is a simple but powerful non-positive curvature property introduced by Kar, roughly requiring that the CAT(0) inequality holds up to sublinear error in the size of the triangle.

We use the asymptotically CAT(0) property to construct visual compactifications for colorable HHGs that provide Z-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit Z-structures. As an application, we prove that many HHGs satisfy the Farrell-Jones Conjecture, providing a new proof for mapping class groups (Bartels-Bestvina) and establishing the conjecture for extra-large type Artin groups.

To construct asymptotically CAT(0) metrics, we show that every colorable HHG admits a manifold-like family of local approximations by CAT(0) cube complexes, where transition maps are cubical almost-isomorphisms.

Abstract: We give geometric conditions which imply that the space obtained by coning off the boundary components of a hyperbolic manifold $M$ is negatively curved. Moreover, we give explicit geometric conditions under which a locally convex subset of $M$ gives rise to a locally convex subset of the cone-off. Group-theoretically, we conclude that the fundamental group of the cone-off is hyperbolic of cohomological dimension $n$ and the $\pi_1$-image of the coned-off locally convex subset is a quasi-convex subgroup. This is joint work with Jason Manning.

Abstract: A classical theorem of Hopf and Freudenthal states that if G is a finitely generated group, then the number of ends of G is either 0, 1, 2 or infinity. We prove a higher-dimensional analogue of this result, showing that if F is a field, G is countable, and Hk(G,FG)=0 for k<n, then dim Hn(G,FG)=0,1 or ∞, significantly extending work of Farrell from 1975. Moreover, in the case dim Hn(G,FG)=1, then G must be a coarse Poincaré duality group. We prove an analogous result for metric spaces. In this talk, we talk about the tools needed to prove this result. We will introduce several coarse topological invariants of metric spaces, inspired by group cohomology. We define the coarse cohomological dimension of a metric space, and demonstrate that if G is a countable group equipped with a proper left-invariant metric, then the coarse cohomological dimension of G coincides with its cohomological dimension whenever the latter is finite. Extending a result of Sauer, we show that coarse cohomological dimension is invariant under coarse equivalence. We characterize unbounded quasi-trees as quasi-geodesic metric spaces of coarse cohomological dimension one.

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