Our seminar meets Tuesdays at 4:30 in JCC 502. Talks are in person and also streamed on Zoom. You can subscribe to the GGTT mailing list at https://elist.tufts.edu/sympa/info/ggtt for details and announcements. The seminar organizers are Kim Ruane, Lorenzo Ruffoni and Genevieve Walsh.

Jan 24Lorenzo Ruffoni (Tufts)Hyperbolization, cubulation, and applications
Abstract:  Gromov introduced some “hyperbolization” procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature, and Ontaneda provided a Riemannian version that works in the smooth setting. These procedures have been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. In joint work with J. Lafont, we construct actions of the fundamental groups of these spaces on CAT(0) cube complexes. As an application, we obtain that these groups are virtually special, hence linear over the integers and residually finite. In particular, we obtain new examples of negatively curved Riemannian manifolds whose fundamental groups virtually algebraically fiber.
Feb 07Chandrika Sadanand (Bowdoin)Hyperbolic cone surfaces and polygonal billiards
Abstract: Consider a polygon-shaped billiard table in the hyperbolic plane on which a ball can roll along geodesics and reflect off of edges infinitely. In joint work with Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of the polygon and the set of possible sequences of edges visited consecutively by billiard balls rolling and reflecting around the polygon. In order to do this, we made an arguably more interesting characterization: when a hyperbolic metric with cone points on a surface is determined by the geodesics that do not pass through cone points. In this talk, we will explore these characterizations and the tools used to prove them.
Mar 02 (Thursday)Michael Hull (UNC Greensboro)Random walks and convex cocompactness in acylindrically hyperbolic groups
Abstract: For a group G acting acylindrically on a hyperbolic metric space X, we say a subgroup H is convex cocompact if the orbit map quasi-isometrically embeds H into X. We study how the elements produced by random walks on G interact with these convex cocompact subgroups. In particular, we show that the subgroup generated by H and a random element is a convex cocompact free product and that the action of G on the space of all infinite index convex cocompact subgroups is topologically transitive. This is joint work with C. Abbott, A. Minasyan, and D. Osin.
Mar 07Tina Torkaman (Harvard)Intersection number and intersection points of closed geodesics on hyperbolic surfaces
Abstract: In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.
Mar 14Carolyn Abbott (Brandeis)(CANCELLED) Morse boundaries of CAT(0) cube complexes
Abstract: The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory. In particular, there is a well- defined notion of the visual boundary of a hyperbolic group. When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant. Instead, one can consider a certain subspace of the visual boundary, called the (sublinearly) Morse boundary. In this talk, I will describe a new topology on this boundary and use it to show that the Morse boundary with the restriction of the visual topology is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes. This result is in contrast to Cashen’s result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general. This is joint work with Merlin Incerti-Medici.
Mar 21spring recess
Apr 04Subhadip Dey (Yale) TBA
Abstract: TBA
Apr 11Alex Suciu (Northeastern)TBA
Abstract: TBA
Apr 25Becca Winarski (Holy Cross) Thurston theory: unifying dynamical and topological
Abstract: Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. Moreover, our proof gives a new framework for classifying self-covering spaces of the torus and Lattés maps. This is joint work with Jim Belk and Dan Margalit.