Past Seminars › Seminar Series

Junior Geometric Group Theory Seminar (past)

Wed, 15/05
16:00 		Martin Finn-Sell (University of Southampton) Junior Geometric Group Theory Seminar Add to calendar SR2
Partial actions of Groups in Coarse Geometry
Group actions play an important role in both topological problems and coarse geometric conjectures. I will introduce the idea of a partial action of a group on a metric space and explain, in the case of certain classes of coarsely disconnected spaces, how partial actions can be used to give a geometric proof of a result of Willett and Yu concerning the coarse Baum-Connes conjecture.
Wed, 08/05
16:00 		David Hume (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Amenable hyperbolic groups
The integers (while wonderful in many others respects) do not make for fascinating Geometric Group Theory. They are, however, essentially the only infinite finitely generated group which is both hyperbolic and amenable. In the class of locally compact topological groups, the intersection of these two notions is richer, and the major aim of this talk will be to give the structure of a classification of such groups due to Caprace-de Cornulier-Monod-Tessera, beginning with Milnor's proof that any connected Lie group admitting a left-invariant negatively curved Riemannian metric is necessarily soluble.
Wed, 01/05
16:00 		Robert Kropholler (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Some Decision Problems in Groups


To continue the day's questions of how complex groups can be I will be looking about some decision problems. I will prove that certain properties of finitely presented groups are undecidable. These properties are called Markov properties and include many nice properties one may want a group to have. I will also hopefully go into an algorithm of Whitehead on deciding if a set of n words generates F_n.
Wed, 24/04
16:00 		(Various) (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Open Problems Session
Wed, 06/03
16:00 		Henry Bradford (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
From Riches to RAAGs: Special Cube Complexes and the Virtual Haken Theorem (Part 2)
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.
Wed, 20/02
16:00 		Alejandra Garrido Angulo (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Self-similar groups
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.
Wed, 13/02
16:00 		Martin Palmer (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Configuration spaces and homological stability (or, what I did for the last three and a half years)
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.
Wed, 06/02
16:00 		Montserrat Casals-Ruiz (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Equations over groups
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.
Wed, 30/01
16:00 		David Hume; Robert Kropholler; Martin Palmer and Alessandro Sisto (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Uniform Hyperbolicity of the Curve Graph
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.
Wed, 30/01
12:00 		Lukasz Grabowski (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR1
Outomorphisms of Out(F_n) are trivial for n>2
The eponymous result is due to Bridson and Vogtmann, and was proven in their paper "Automorphisms of Automorphism Groups of Free Groups" (Journal of Algebra 229). While I'll remind you all the basic definitions, it would be very helpful to be already somewhat familiar with the outer space.
Wed, 23/01
16:00 		John Mackay (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Cubulating small cancellation and random groups
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).
Wed, 16/01
16:00 		Robert Kropholler (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Relations between some topological and group theoretic conjectures
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.
Mon, 03/12/2012
00:00 		Diana Davis (Brown University) Junior Geometric Group Theory Seminar Add to calendar SR2
Cutting sequences and Bouw-Möller surfaces

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.
Wed, 28/11/2012
16:00 		Will Cavendish (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Engulfed subgroups of discrete groups

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.
Wed, 21/11/2012
16:00 		Andrew Sale (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Magnus QI: the motion picture, featuring the Magnus embedding
Let F be a free group, and N a normal subgroup of F with derived subgroup N'. The Magnus embedding gives a way of seeing F/N' as a subgroup of a wreath product of a free abelian group over over F/N. The aim is to show that the Magnus embedding is a quasi-isometric embedding (hence "Q.I." in the title). For this I will use an alternative geometric definition of the embedding (hence "picture"), which I will show is equivalent to the definition which uses Fox calculus. Please note that we will assume no prior knowledge of calculus.
Thu, 15/11/2012
16:30 		Søren Fuglede Jørgensen (Aarhus University) Junior Geometric Group Theory Seminar Add to calendar St John's
Quantum representations and their algebraic properties
In this part, I will redefine the quantum representations for $ G = SU(2) $ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial. Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.
Wed, 31/10/2012
16:00 		Jason Semeraro Junior Geometric Group Theory Seminar Add to calendar SR2
Bass-Serre Theory for Fusion Systems
Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.
Wed, 24/10/2012
16:00 		David Hume Junior Geometric Group Theory Seminar Add to calendar L2
The metric geometry of mapping class groups
We give a brief overview of hyperbolic metric spaces and the relatively hyperbolic counterparts, with particular emphasis on the quasi-isometry class of trees. We then show that an understanding of the relative version of such spaces - quasi tree-graded spaces -  has strong consequences for mapping class groups. In particular, they are shown to embed into a finite product of (possibly infinite valence) simplicial trees. This uses and extends the work of Bestvina, Bromberg and Fujiwara.
Wed, 17/10/2012
16:00 		Elisabeth Fink (University of Oxford) Junior Geometric Group Theory Seminar Add to calendar SR2
Words and growth of groups acting on rooted trees
I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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