Daily Schedule
All talks will be held in Exley Science Center, Room 113
Wednesday/Thursday
Friday
Talk information (Titles and Abstracts)
Manisha Garg "Boundaries at infinity"
Boundaries at infinity translate the large-scale geometry of spaces and groups into the geometry of directions of escape. We will begin with the Gromov boundary of a hyperbolic space, study several topological and metric properties can reflect algebraic and coarse-geometric properties of the group. We will then explore what happens beyond hyperbolicity, introducing Morse and sublinearly Morse boundaries through the example of a CAT(0) cubical tree of flats. In the second part, we will discuss concrete metric and analytic structures on boundaries, including visual metrics along with various other metrics, quasisymmetric geometry, and various dimensions. Our goal is to illustrate how analytical and measure-theoretic tools can encode information about the large-scale geometry of the ambient space or group.
Rachmiel Klein "Coxeter groups' elegant actions and divergence"
In the first lecture, we will construct the Davis complex, the natural CAT(0) space that Coxeter groups act on, and give examples. The Davis complex also provides many examples of cubulated groups, because in the case of right angled Coxeter groups, it is a CAT(0) cube complex. In the second lecture, we introduce the QI invariant "divergence," which is closely related to non-positive curvature, and discuss divergence results for right angled Coxeter groups.
Changqian Li "Anti-tori in products of trees: aperiodicity, fractals, and virtual specialness"
An anti-torus is a non-periodic flat in a product of two trees whose horizontal and vertical axes are periodic. Introduced by Dani Wise, anti-tori give a geometric criterion for irreducibility and provide an obstruction to virtual specialness for complete square complexes. I will first explain Wise’s construction of an anti-torus, in which non-periodicity arises from a period-doubling phenomenon. I will then discuss the recent construction of fractal anti-tori by Caprace and Vast, whose square tilings are generated by self-similar substitution rules. These examples reveal close connections among irreducible lattices in products of trees, aperiodic tilings, automata, and virtual specialness.
Sam Shepherd "Commensurable groups, common finite covers, and separable subgroups"
If a pair of compact length spaces have a common universal cover, do they necessarily have a common finite-sheeted cover? We will discuss answers to this question for various different classes of spaces, some in a cubical setting and some not. This question is closely connected to group theory because the existence of a common finite-sheeted cover implies that the fundamental groups of the spaces are commensurable. A particular focus will be given to my result about the commensurability of lattices in right-angled buildings. One of the key ingedients for my result is the separability of certain subgroups; we will explore how the separability of subgroups is used in the proof of my theorem, and also why it is an important concept in a wider context - especially regarding the theory of special cube complexes.
Jagerynn Verano "The Sageev construction"
This talk focuses on connections between the subgroup structure of a group and the types of actions that it admits on CAT(0) cube complexes. Group actions on CAT(0) cube complexes arise from Sageev's construction, which takes as input a collection of codimension-one subgroups and outputs an isometric action with no global fixed point. On the one hand, one may ask for structural conditions under which ''nice" (such as proper or cocompact) actions are obtained. On the other hand, one may be interested in structural results obtained from different kinds of actions. In this talk, we will discuss both of these questions, with an emphasis on the setting of cocompact actions.