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.