Title: Algebraic conditions for characterizing hierarchically hyperbolic groups
Abstract: Hierarchical hyperbolicity was developed to capture key geometric features of the mapping class group of a surface, resulting in a framework for dissecting some other not-quite-hyperbolic groups into neatly arranged collections of hyperbolic pieces. This analogy with mapping class groups is lucrative, albeit sometimes cumbersome. This talk focuses on a comprehensive algebraic characterization of when a hierarchically hyperbolic structure exists for a free-by-cyclic group, using a straightforward condition on intersections between particular maximal subgroups. This represents joint work with Mark Hagen, Funda Gültepe, and Pritam Ghosh.
If time permits, I will discuss how this algebraic perspective and related results inform my current work in progress (joint with Alex Wright) on whether surface braid groups are hierarchically hyperbolic.
The talk is intended to be self-contained and accessible to an audience familiar with group actions. Knowledge of
hyperbolic geometry will not be assumed, although foundational facts will be presented without proof.
THOMAS KOBERDA
JOHANNA MANGAHAS
Abstract: Divisible convex sets have long been important in the study of Hilbert geometries. When a divisible convex set is an ellipsoid, the Hilbert geometry
it induces is the hyperbolic space. Strictly convex divisible domains exhibit some nonpositive curvature properties, but only the ellipsoid is a CAT(0) space.
The notion of p-uniform convexity from the theory of Banach spaces has been proposed by Shin-ichi Ohta as a generalization of the Alexandrov-Toponogov
comparison theorems to Finsler manifolds. In this talk, I will give an overview of strictly convex divisible domains and discuss how the regularity of their
boundary implies that the natural Finsler metric they can be equipped with is $\beta$-uniformly convex.
INTRODUCTORY TALK : What is Hierarchical Hyperbolicity?
Abstract: Hierarchical hyperbolicity is a powerful framework that provides deep geometric information about groups and spaces
that have particularly nice decompositions in a collection of hyperbolic spaces. Examples include a wide variety of objects from
the theory of mapping class groups, many cubical groups, and large classes of Artin groups. We will provide a gentle
introduction to hierarchical hyperbolicity through the lens of a couple of concrete and transparent examples.
TALK II: Non-proper bundles and geometric finiteness in the mapping class groups
Abstract: In theory, there is an amazing dictionary between the properties of surface group extensions and homomorphisms into the mapping class group.
In practice, this dictionary is poorly understood with little known about how properties of the extension groups correspond to maps into the mapping class
group. Several particularly nice examples suggest that hierarchical hyperbolicity of surface group extensions should correspond to some sort of "geometric
finiteness" for subgroups of the mapping class group. The primary obstruction to extrapolating a general theory from these nice examples has been the lack of a combination theorem for bundles of hyperbolic graphs where the fibers are not properly embedded. We will give an overview of the state of this theory and then present the first combination theorem for these bundles. As an application, we prove the hierarchical hyperbolicity of the simplest class of "geometrically finite" subgroups. Joint work with Spencer Dowdall, Matt Durham, Chris Leininger, and Alex Sisto.
Title: Polygonal Billiards with obstacles: “hearing” homotopy
Abstract: Consider a polygon-shaped billiard table with some heavy, polygonal obstacles placed on top. A ball can roll along straight lines and reflect off of edges (of the table and of the obstacles) infinitely. In forthcoming work, joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table (including the position and the shape of the polygonal obstacles!) and the set of possible infinite edge itineraries of balls travelling on it. To solve this problem, one has to figure out how a billiard path navigates topologically around the obstacles between each pair of consecutive bounces, when given just the list edge itineraries.
Abstract: In their paper, Branman, Domat, Hoganson and Lymann proved that if a topological group acts in a "nice" way on a simplicial graph, then the group has a well defined geometry that makes it quasi-isometric to the graph. These actions generalize a Svârc-Milnor action to the context of coarsely boundedly (CB) generated Polish groups.
We adapt these ideas to the context of locally bounded Polish groups and then construct an arc and curve model coarsely equivalent to Map(S) when Map(S) is locally bounded and S has a non-displaceable subsurface. We then use this model to show that the asymptotic dimension of Map(S) is infinite.
Title: Gromov Boundary of the Grand Arc Graph.
Abstract: In 1999, E. Klarreich found a very intriguing correspondence between the Gromov boundary of the curve graph for closed surfaces (a very GGT object) with the space of ending laminations on the surface (a very geometric object). Since then, Hamendstädt, Schleimer and Pho-On have thought about various proofs for this result, and generalizations to the arc graph / the arc-and-curve graph for finite-type surfaces. The grand arc graph is a type of arc graph associated with certain infinite-type surfaces, which is also an infinite-diameter hyperbolic graph. In this talk, we shall talk about a couple of ways to define “laminations that should correspond to points on the Gromov boundary of the grand arc graph”. This work is joint with Carolyn Abbott and Assaf Bar-Natan.
INTRODUCTORY TALK: Commutator techniques in transformation groups
Abstract: Throughout the literature on the algebraic and geometric structure of homeomorphism groups of manifolds, commutators play a central role in arguments. The goal of this talk is to illustrate a few of these techniques; to do so, we will prove that the connected component of the identity in the homeomorphism group of a closed manifold is a simple group. Time permitting, we will also use commutators to exhibit a strong form of distortion in these groups for homeomorphisms supported in balls.
TALK II: Strongly bounded generation in transformation groups
Abstract: Up to quasi-isometry, finitely generated groups admit a canonical left-invariant metric, making coarse-geometric invariants into group-theoretic invariants. Can non-finitely generated groups have a well-defined coarse geometry in the same sense? In this talk, we exhibit such examples by "going against nature" — forgetting the topology of several large, rich families of topological transformation groups (e.g., homeomorphism groups of manifolds) and showing that they nevertheless admit canonical large-scale geometries as abstract groups.
Title: Counting surface subgroups in cusped hyperbolic 3-manifolds and atoroidal surface bundles
Abstract:
The recent discovery of atoroidal surface bundles by Kent and Leininger is an important breakthrough in geometric group theory. In their paper, they find
infinitely many commensurability classes of purely pseudo-Anosov surface subgroups in some mapping class group.
Our recent work counts quasi-Fuchsian surfaces of genus at most g in a cusped hyperbolic 3-manifold M. Up to conjugacy and commensurability, we show
this count grows on the order of (cg)^2g for some constant c depending on M. This result extends the work of Kahn-Markovic for a closed hyperbolic 3-
manifold and establishes a lower bound in terms of the genus for the commensurability classes of Kent-Leininger surface bundles.
In contrast, for some fixed genus of at least 2, the upper bound does not exist for the surface subgroups with accidental parabolics up to conjugacy; we
demonstrate this by explicitly constructing infinitely many such examples. This is joint work with Xiaolong Hans Han and Zhenghao Rao.
