Past Junior Topology and Group Theory Seminar

30 October 2019
16:00
Naya Yerolemou
Abstract

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

  • Junior Topology and Group Theory Seminar
23 October 2019
16:00
Harry Petyt
Abstract

The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. The talk will be focused on a way to understand such a group by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". We'll finish with a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.

  • Junior Topology and Group Theory Seminar
16 October 2019
16:00
Abstract

Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which leads to the natural question of if there exist any vertex-transitive graphs that are completely unlike any Cayley graph. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.

 

  • Junior Topology and Group Theory Seminar
19 June 2019
16:00
Nicolaus Heuer
Abstract

Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. 

I will give an introduction to simplicial volume and describe a recent result with Clara Löh (University of Regensburg), showing that the set of simplicial volumes in higher dimensions is dense in $R^+$.

  • Junior Topology and Group Theory Seminar
29 May 2019
16:00
Sam Shepherd
Abstract

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will present a new proof of this using groupoids, and then talk about two generalisations of the theorem that can also be tackled with this groupoid approach: one gives us control over the local structure of the common finite cover, and the other deals with graphs of spaces.

  • Junior Topology and Group Theory Seminar
22 May 2019
16:00
Abstract

The last decade or so has seen substantial progress in the theory of (outer) automorphism groups of right-angled Artin groups (RAAGs), spearheaded by work of Charney and Vogtmann. Many of the techniques used for RAAGs also apply to a wider class of groups, graph products of finitely generated abelian groups, which includes right-angled Coxeter groups (RACGs). In this talk, I will give an introduction to automorphism groups of such graph products, and describe recent developments surrounding the outer automorphism groups of RACGs, explaining the links to what we know in the RAAG case.

  • Junior Topology and Group Theory Seminar
15 May 2019
16:00
Michal Buran
Abstract


It is often fruitful to study an infinite discrete group via its finite quotients.  For this reason, conditions that guarantee many finite quotients can be useful.  One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.

  • Junior Topology and Group Theory Seminar
10 May 2019
13:00
Abstract

The discrete fundamental groups of a metric space can be thought of as fundamental groups that `ignore' closed loops up to some specified size R. As the parameter R grows, these groups have been used to produce interesting invariants of coarse geometry. On the other hand, as R gets smaller one would expect to retrieve the usual fundamental group as a limit. In this talk I will try to briefly illustrate both these aspects.

  • Junior Topology and Group Theory Seminar

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