From Riches to RAAGs: Special Cube Complexes and the Virtual Haken Theorem (Part 2)
Abstract
I will outline Bergeron-Wise’s proof that the Virtual Haken Conjecture follows from Wise’s Conjecture on virtual specialness of non-positively curved cube complexes. If time permits, I will sketch some highlights from the proof of Wise’s Conjecture due to Agol and based on the Weak Separation Theorem of Agol-Groves-Manning.
Self-similar groups
Abstract
Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.
Configuration spaces and homological stability (or, what I did for the last three and a half years)
Abstract
First of all, I will give an overview of what the
phenomenon of homological stability is and why it's useful, with plenty
of examples. I will then introduce configuration spaces -- of various
different kinds -- and give an overview of what is known about their
homological stability properties. A "configuration" here can be more
than just a finite collection of points in a background space: in
particular, the points may be equipped with a certain non-local
structure (an "orientation"), or one can consider unlinked
embedded copies of a fixed manifold instead of just points. If by some
miracle time permits, I may also say something about homological
stability with local coefficients, in general and in particular for
configuration spaces.
Equations over groups
Abstract
The theory of equations
over groups goes back to the very beginning of group theory and is
linked to many deep problems in mathematics, such as the Diophantine
problem over rationals. In this talk, we shall survey some of the key
results on equations over groups,
give an outline of the Makanin-Razborov process (an algorithm for
solving equations over free groups) and its connections to other results
in group theory and low-dimensional topology.
Uniform Hyperbolicity of the Curve Graph
Abstract
We will discuss (very) recent work by Hensel; Przytycki and Webb, who describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
Cubulating small cancellation and random groups
Abstract
I'll discuss work of Wise and Ollivier-Wise that gives cubulations of certain small cancellation and random groups, which in turn shows that they do not have property (T).
Relations between some topological and group theoretic conjectures
Abstract
I will be looking at some conjectures and theorems closely related to the h-cobordism theorem and will try to show some connections between them and some group theoretic conjectures.
00:00
Cutting sequences and Bouw-Möller surfaces
Abstract
We will start with the square torus, move on to all regular polygons, and then look at a large family of flat surfaces called Bouw-Möller surfaces, made by gluing together many polygons. On each surface, we will consider the action of a certain shearing action on geodesic paths on the surface, and a certain corresponding sequence.