Spring 2023

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.

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.

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.

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.

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.

Abstract: The classical Klein Combination Theorem provides sufficient conditions to construct new Kleinian groups. Subsequently, Maskit gave far-reaching generalizations to the Klein Combination Theorem. A special feature of Maskit’s theorems is that they furnish sufficient conditions so that the combined group retains nice geometric features, such as convex-cocompactness or geometric-finiteness. In recent years, Anosov subgroups have emerged as a natural higher-rank generalization of the convex-cocompact Kleinian groups, exhibiting their robust geometric and dynamical properties. This talk will discuss my joint work with Michael Kapovich on Combination Theorems in the setting of Anosov subgroups.

Abstract: There are several topological invariants associated to
a finite-type CW-complex, which keep track of various finiteness
properties of its covering spaces. These invariants, which include
the cohomology jump loci and the BNSR invariants, are interconnected
in ways that makes them both more computable and more informative.
I will describe one such connection, made possible by tropical geometry,
and I will provide examples and applications pertaining to group theory,
low-dimensional topology, and complex geometry.

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.