ELIOT BONGIOVANNI
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
INTRODUCTORY TALK: WHAT IS GEOMETRIC GROUP THEORY?
I will give an introduction to geometric group theory and where it falls within the broader context of mathematics.
TALK II: Using logic in geometric group theory
Abstract: I will discuss some recent work which highlights the applications of logic to mapping class groups, right-angled Artin
groups, and homeomorphism groups of manifolds.
JOHANNA MANGAHAS
Title: Recognizing Stable Subgroups of Graph Products
Abstract: Stability for subgroups of finitely generated groups generalizes quasiconvexity for subgroups of hyperbolic groups: stable subgroups are quasi-isometrically embedded, and ambient-group quasi-geodesics between points in the subgroup fellow travel each other (from which it follows that the subgroup is hyperbolic, whereas the ambient group generally is not). Since its definition by Durham and Taylor in 2015, subgroup stability has been investigated for many kinds of groups. In this talk, we consider the question of whether/when a subgroup’s stability is recognized by some action of the larger group on some hyperbolic space, by which we mean that the subgroup quasi-isometrically embeds into the space by the orbit map. Right-angled Artin groups (as well. as mapping class groups, and more generally hierarchically hyperbolic groups) each have a well-understood hyperbolic space that recognizes all their stable subgroups; we call this a universal recognizing space. Replace the role of vertex generators for a RAAG with arbitrary vertex groups, and one has the more general notion of a graph product. We prove that universal recognizing spaces exist for graph products with finitely generated, infinite vertex groups, when the underlying graph has no isolated vertices. Moreover we extend to these graph products previous characterizations of subgroup stability in RAAGs. This is joint work with Sahana Balasubramanya, Marissa Chesser, Alice Kerr, and Marie Trin.
AMELIA POMPILIO
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.
JACOB RUSSELL
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.
CHANDRIKA SADANAND
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.
GEORGE SHAJI
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.
ARYA VADNERE
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.
NICK VLAMIS
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.
JIA WAN
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.
Ruben Mamani, University of Toledo
Title: Center and derivations of U(sl2) over Z/p^nZ.
Abstract: Let A be the universal enveloping algebra U(sl2) over Z/p^n Z for an odd prime p. We describe the center of A: it is generated by the Casimir element Delta over the image of a canonical embedding from the ring of Witt vectors of length n of the p-center. We also classify all derivations of A and compute the first Hochschild cohomology HH^1(A). In particular, we show that the restriction map from HH^1(A) to the derivations of Z(A) is an isomorphism when n=1 but fails for n>1.
Zhihao Mu, CUNY Graduate Center
Title: Connectivity of Gromov boundary of the maximized hyperbolic space of a right-angled Coxeter group
Abstract: Right-angled Coxeter groups (RACGs) are hierarchically hyperbolic groups(HHGs), and their maximized hyperbolic space is obtained from its Cayley graph by coning-off standard product regions. For the mapping class group, the maximized hyperbolic space is quasi-isometric to the curve graph, whose Gromov boundary is connected (Leininger–Schleimer, Gabai) and even linearly connected (Wright) when the complexity of the surface is at least 2. In this talk, we will give a necessary and sufficient condition on the defining graph of a RACG under which the Gromov boundary of its maximized hyperbolic space is connected and linearly connected. As an application, splitting of a one-ended RACG over an unbounded product subgroups is a quasi-isometric invariant.
Neelam, University of Utah
Abstract: Grigorchuk and Machi introduced a group Γ of intermediate growth that is a subgroup of Aut(R). They do this by showing that the group is left- orderable. In 2008, Andres Navas gave an explicit description of a group G of homeomorphisms of [0, 1]. In his paper, he mentions that G is isomorphic to the geometric realization of Γ. In this talk, I prove this claim by showing that Γ is isomorphic to G. In addition, we see that these groups are isomorphic to the braided Grigorchuk group defined by Skipper and Zaremsky in 2023 which has recently appeared in the study of big mapping class groups by Daniel Alcock . More broadly, this construction of braiding Grigorchuk group fits into a program relating self-similar groups to their braided analogs. As an example of this, I will discuss a general theorem where the amenability property gets preserved under braiding.
Ekaterina Rybak, Vanderbilt
Title: Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups.
Abstract: A group $G$ is mixed identity free (MIF) if there is no nontrivial word $w \in G * F_n$ such that every homomorphism $G * F_n \longrightarrow G$ that is the identity on $G$ sends $w$ to $1$. The transitivity degree of a group $G$ is the maximum $k$ such that a group $G$ admits a faithful $k$-transitive action. For groups admitting general-type action on hyperbolic spaces, a pure algebraic condition of being MIF is equivalent to a dynamical condition of an action on the limit set in the Gromov boundary. I will discuss this result, its corollaries, and mention the application to the transitivity degree of weakly hyperbolic groups.
Aditya De Saha, University of Florida
Abstract: L. Lusternik and L. Schnirelmann introduced their 'category' (now the so-called LS-Category) in 1934 as a numerical invariant to study (among other things) criitcal points of real-valued functions on manifolds. Since then it's been generalized and used in several different settings. We try to do the same in the context of large-scale geometry, defining the 'Coarse LS-Category' (or c-cat in short).
Gerard Thompson, University of Toledo
Title: Invariant Einstein pseudo-metrics and Ricci-flat metrics on four-dimensional Lie groups.
Abstract: I will state a few results