Schedule for: 24w5254 - Advances in Hierarchical Hyperbolicity

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will discuss a criterion of combinatorial flavor to check that a given space is an HHS, in particular giving a "user's guide" on how to apply it in practice. This criterion is quite useful to construct new examples of HHSs, and it is often a lot simpler than checking the axioms directly.

Behrstock-Hagen-Sisto's cubical model theorem says that the hierarchical/median hull of a finite set of points in an HHS admits a quasi-median quasi-isometry to a CAT(0) cube complex. In the first part of the talk, I will describe an alternative construction which realizes the cubical model as a weakly convex subset of a product of simplicial trees naturally arising from the ambient HHS structure. This construction also works for modeling hulls of finitely-many hierarchy rays and points in the HHS boundary. In the second part of the talk, I'll explain how the hierarchical hull of any finite collection of simplices in the HHS boundary is quasi-median quasi-isometric to a CAT(0) cube complex, with the cubical model map extending to a simplicial isomorphism from the simplicial boundary of the cube complex to the given collection of simplices in the HHS boundary.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Mangahas showed that there exists an integer m determined by the underlying finite type surface S so that for every finite generating set A of the mapping class group of S, there is a Morse element in the ball of radius m about the identity. A more fine-tuned question is whether there is a uniform integer m so that for any finite generating set A, *every ball* (not necessarily centered about e) of radius m in Cay(G,A) contains a Morse element; we refer to this as uniform density of Morse elements. I will discuss some recent work showing that both of the aforementioned statements hold for virtually torsion-free HHGs; the fine-tuned statement above is new even for mapping class groups (of finite type surfaces) and is obtained via joint work with Rafi.

Sageev's construction says that, under a finiteness condition, you can make a cube complex dual to a collection of walls by counting how many walls separate each pair of points. With Abdul Zalloum, we observed that if you restrict which families of walls you count, then you can relax the finiteness condition. For an HHS X, this lets you use natural walls (those used in the cubulation of hulls) to build a dual median algebra quasiisometric to X, which has improved fine properties.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss joint work with Autumn Kent in which we construct the first known examples of compact atoroidal surface bundles over surfaces for which the base and fiber genus are both at least 2.

Going beyond the setting of convex cocompactness, there is an effort to develop a theory of geometric finiteness for subgroups of mapping class groups that captures a broader range of behaviors and relates these to the structure of Teichmüller space, the action on the curve complex and the geometry of surface group extensions as viewed, for example, via hierarchical hyperbolicity. I will survey some progress in this direction and highlight various examples that fit into this picture. We will focus on the case of lattice Veech groups, which are perhaps the prototypical candidates for geometric finiteness. I will describe the geometric structure of these groups and how it gives rise to hierarchical hyperbolicity and quasi-isometric rigidity for the associated surface group extensions. Joint work with Matt Durham, Chris Leininger, and Alex Sisto.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given a closed surface $S$, a subgroup $G$ of the mapping class group of $S$ has an associated extension group $\Gamma$, which is the fundamental group of an S-bundle with monodromy an isomorphism to G. A general problem is to infer features of $\Gamma$ from $G$. I take $G$ to be a finitely generated Veech group and show that $Gamma$ is hierarchically hyperbolic. This is a generalization of results from Dowdall, Durham, Leininger, and Sisto regarding lattice Veech groups. The focus of this talk is constructing a hyperbolic space $\hat{E}$ on which $\Gamma$ acts nicely (isometrically and cocompactly). This example contributes to the growing evidence of a good notion of “geometric finiteness” for subgroups of mapping class groups.

We will discuss the parabolically geometrically finite (PGF) subgroups of mapping class groups, which is a class of groups generalizing the definition of convex cocompact groups via the curve complex. This class contains all finitely generated Veech groups, as well as free products of multitwist groups on sufficiently far apart multicurves. We will discuss two results about these groups. First, they are undistorted as subgroups of the mapping class group. Second, we give a combination theorem which allows one to build many more examples of PGF groups. With whatever time is remaining afterwards, we will discuss open problems.

A result of Behrstock, Hagen, and Sisto states that, under suitable conditions, a self-quasi-isometry of a hierarchically hyperbolic space induces an automorphism of the "hinge graph", which roughly encodes the intersection patterns of standard quasiflats of the maximum dimension. If one is able to identify who this graph is, and most importantly the group of its simplicial automorphisms, then one can aim at some classification of quasi-isometries. In this talk we will first review the construction of the hinge graph and its automorphism. The machinery can then be applied to prove that random quotients of mapping class groups of surfaces are quasi-isometrically rigid, meaning that if such a quotient and a finitely generated group are quasi-isometric then they are weakly commensurable. The key point is that, in this case, the hinge graph is related to the corresponding quotient of the curve graph, whose automorphism group is the quotient group itself (this is the analogue of a result of Ivanov-Korkmaz for mapping class groups). If time permits, we will also show how quasi-isometric rigidity can be used to deduce other "rigidity" results, such as the fact that all automorphisms of such quotients are inner.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk aims to explain the state of the art of hyperbolicity in Artin groups and related complexes. We will see what the latest advances have been in hyperbolicity in the sense of Gromov, acylindrical hyperbolicity, and hierarchical hyperbolicity, as well as the main techniques that have been used.

Exponential growth is well-known to be preserved under quasi-isometry. While, changing finite generating sets may yield exponential growth rate that are not uniformly bounded over all finite generating sets, uniform exponential growth has been proven for RAAGs using algebraic tools, and for mapping class groups using actions on homology or curve complexes. One of the challenges in determining uniform exponential growth in acylindrically hyperbolic group is controlling elliptic subgroups. I will discuss joint work with Abbott and Spriano that constrains dynamic of elliptic subgroups on lower level domains using the hierarchy structure. These tools give a unified geometric approach for showing uniform exponential growth in HHGs and have seen various other applications in hierarchically hyperbolic groups.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

There has been a great deal of work on median spaces and groups acting on them; these spaces are natural and important, being a common generalisation of real trees and CAT(0) cube complexes. Another natural source of median algebras/median metric spaces comes from large-scale geometry, namely work by Bowditch and Zeidler showing that asymptotic cones of finite-rank coarse median spaces are canonically bilipschitz to median spaces. In particular, studying asymptotic cones of HHGs means studying median spaces. But in exactly the same way that passing from a cube complex to an HHS structure requires a "descending chain condition" on certain convex subcomplexes --- namely the existence of a factor system --- there is a subclass of connected, finite-rank median spaces that admit a factor system in an appropriate continuous sense; these we called "real cubings" and have various advantageous properties compared to general median spaces. These relate back to hierarchical hyperbolicity via our first result: asymptotic cones of HHSes are bilipschitz to real cubings. This extends an earlier result of Behrstock-Drutu-Sapir on mapping class groups. The next step is the construction, inspired by related work of Osin-Sapir and Sisto for tree-graded spaces, is to construct so-called universal real cubings determined by local data. An interesting feature of universal real cubings is that they are not only spaces, but groups. I will explain the sense in which they are "continuous RAAGs". I will then briefly outline how these pieces fit into a proof that many hierarchically hyperbolic groups have unique asymptotic cones (up to bilipschitz equivalence). This is all joint work with Montserrat Casals-Ruiz and Ilya Kazachkov.

For a vertex c and an integer radius $r$, the sphere $S_r(c)$ is the induced graph on the set of vertices of distance $r$ from $c$. We will show that spheres in the curve graph are typically connected, and discuss connectivity properties of the Gromov boundary. We will also explain the motivation and context for this work.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Uniform exponential growth of finitely generated groups has been a classical problem in geometric group theory. In recent years, there is an increasing interest in understanding the product set growth, which could be thought of as a stronger version of uniform exponential growth. In this talk, I will discuss these two problems, provided that the finitely generated group under consideration acts properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are full-filled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups. Via a weakly acylindrical action on each factor, we are able to classify subgroups with product set growth in any group acting discretely on a simply connected manifold with pinched negative curvature, and in groups acting acylindrically on trees, and in 3-manifold groups. This is based on a joint work with Renxing Wan (PKU).

We shall start with a recent theorem (joint with Nir Lazarovich and Alex Margolis) that states the following: Let $H < G$ be an infinite commensurated hyperbolic subgroup of infinite index in a hyperbolic group. Then H is virtually a free product of surface and free groups. This leads us to the study of the abstract commensurator group $\text{Comm}(H)$ of surface groups $H$. The rest of the talk will be a survey of what is known, going back to the work of Sullivan, Biswas, Nag, Penner, Saric and others. It turns out that $\text{Comm}(H)$ naturally acts on spaces that admit a "hierarchically hyperbolic structure with an infinite hierarchy"

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.