Abstracts-2023-cube

Median spaces are a simultaneous generalisation of L^1 spaces, CAT(0) cube complexes (their discrete counterpart) and R-trees (their 1-dimensional counterpart). They naturally occur in two disparate areas of geometric group theory.
On the one hand, finite-dimensional median spaces arise as degenerations of sequences of spaces that coarsely resemble cube complexes. As such, they have applications to the study of quasi-flats, quasi-isometries and automorphisms for broad classes of groups: importantly, those with an HHS structure (such as mapping class groups) and those that can be cocompactly cubulated (such as RAAGs).
At the same time, infinite-dimensional median spaces (and their dual concept: spaces with measured walls) can also be a useful tool to test more “functional analytic” properties for a given group, particularly Kazhdan’s property (T) and the Haagerup property.
I will give an overview of these two aspects of the subject, also focusing on the connections to the theory of median algebras and its implications for the study of cocompactly cubulated groups.

We will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. We will focus on nonpositive features, such as local properties, bicombings and classification of isometries. We will present various rich constructions of such spaces, quite often associated with groups of geometric nature, such as hyperbolic groups, braid groups, mapping class groups and higher rank lattices.

Combinatorial and geometric methods have a long history in group theory, often with connections or parallels to classical forms of nonpositive curvature. In this minicourse we will learn about classical small cancellation theory, its connections to systolic and quadric complexes and how these classes of combinatorially nonpositively curved spaces can be applied to study groups. We will also discuss more general classes of spaces with a view towards unification.

This minicourse will serve as an introduction to CAT(0) cube complexes.

In joint work with Charney and Vogtmann, we constructed an outer space for the automorphism group of a right-angled Artin group (RAAG). This is a contractible, finite-dimensional space on which the automorphism group of the RAAG acts with finite stabilizers. The Nielsen realization problem asks whether every finite subgroup of automorphisms fixes a point under this action. We show that finite subgroups of the untwisted automorphism group of a RAAG satisfy Nielsen realization and discuss a strategy for extending this result to all finite subgroups.

The theory of real trees and groups acting on them has had a deep impact on Group Theory by providing tools to attack new problems, by simplifying proofs of classical results, and by establishing new connections.

Graphs with convex balls have been introduced and characterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). They generalize bridged/systolic graphs that have been introduced in the same papers.
In this work, we give a local-to-global characterization of graphs with convex balls, as graphs in which balls of radius 3 are convex and whose triangle-pentagon complexes are simply connected. We also establish that their Rips complexes are contractible. Finally, we showthat groups acting geometrically on graphs with convex balls arebiautomatic.
Joint work with V. Chepoi and U. Giocanti

In this talk I will explain a joint work with Javier Aramayona, Julio Aroca, Rachel Skipper and Xiaolei Wu. We define a new family of groups that are subgroups of the mapping class group M ap(Σg) of asurface Σg of genus g with a Cantor set removed and we will call these groups Block Mapping ClassGroups B(H), where H is a subgroup of Σg. More visually, this family will be constructed by making a tree-like surface gluing pair of pants and taking homeomorphisms that depend on H with certainpreservation properties (it will preserve what we will call a block decomposition of this surface, hence the name of our groups). We will see that this family is closely related to Thompson’s groups and that
it has the property of being of type Fn if and only if H is. As a consequence, for every g ∈ N ∪ {0, ∞} and every n ≥ 1, we construct a subgroup G < (Σg) that is of type Fn but not of type Fn+1, and which contains the mapping class group of every compact surface of genus less or equal to and with nonempty boundary. As expected in this workshop, the techniques involve manipulating cube complexes, as the Stein-Farley cube complex.

Groves and I introduced relatively geometric actions on CAT(0) cube complexes to study relatively hyperbolic groups that act on CAT(0) cube complexes. In this talk, I will explain how to construct relatively geometric actions using a boundary criterion. I will discuss some of the properties of relatively geometric actions and how we hope to use these tools to develop structure theorems for groups that admit relatively geometric actions. This is primarily joint work with Daniel Groves and may also include joint work with Thomas Ng.