Past Junior Topology and Group Theory Seminar

10 March 2021
10:00
David Sheard
Abstract

In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.

  • Junior Topology and Group Theory Seminar
3 March 2021
10:00
Marco Barberis
Abstract

Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.

During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".

  • Junior Topology and Group Theory Seminar
24 February 2021
10:00
Monika Kudlinska
Abstract

A 3-manifold fibers over the circle if it can be identified with the mapping torus of a surface homeomorphism. If the surface is compact with non-empty boundary then the corresponding 3-manifold group is free-by-cyclic, and the action of the cyclic group on the free group is induced by the surface homeomorphism. Although most free-by-cyclic groups do not arise as fundamental groups of 3-manifolds which fiber over the circle, there is a strong analogy between the two families.

In this talk I will discuss how dynamical properties of the monodromy affect the geometry/algebra of the corresponding mapping torus. We will see how the same 3-manifold or group can admit multiple fiberings and what properties of the monodromy are known to be preserved under different fiberings.

  • Junior Topology and Group Theory Seminar
17 February 2021
10:00
Sam Fisher
Abstract

This talk will be an introduction to L^2 homology, which is roughly "square-summable" homology. We begin by defining the L^2 homology of a G-CW complex (a CW complex with a cellular G-action), and we will discuss some applications of these invariants to group theory and topology. We will then focus on a criterion of Wise, which proves the vanishing of the 2nd L^2 Betti number in combinatorial CW-complexes with elementary methods. If time permits, we will also introduce Wise's energy criterion.
 

  • Junior Topology and Group Theory Seminar
3 February 2021
10:00
Patrick Nairne
Abstract

The asymptotic cone of a metric space X is what you see when you "look at X from infinitely far away". The asymptotic cone therefore captures much of the large scale geometry of the metric space. Furthermore, the construction often produces a smooth space from a discrete one, allowing us to apply the techniques of calculus. Notably, Gromov used asymptotic cones in his proof that finitely generated groups of polynomial growth are virtually nilpotent.

In the talk I will define asymptotic cones using the language of ultrafilters and ultralimits. We will then look at the particular cases of asymptotic cones of virtually nilpotent groups and hyperbolic metric spaces. At the end, we will prove a result of Gromov which relates the fundamental group of the asymptotic cone to the filling order of the underlying metric space.

  • Junior Topology and Group Theory Seminar
27 January 2021
10:00
Adele Jackson
Abstract

A major tool used to understand manifolds is understanding how different measures of complexity relate to one another. One particularly combinatorial measure of the complexity of a 3-manifold M is the minimal number of tetrahedra in a simplicial complex homeomorphic to M, called the triangulation complexity of M. A natural question is whether we can relate this with more geometric measures of the complexity of a manifold, especially understanding these relationships as combinatorial complexity grows.

In the case when the manifold fibres over the circle, a recent theorem of Marc Lackenby and Jessica Purcell gives both an upper and lower bound on the triangulation complexity in terms of a geometric invariant of the gluing map (its translation length in the triangulation graph). We will discuss this result as well as a new result concerning what happens when we alter the gluing map by a Dehn twist.

  • Junior Topology and Group Theory Seminar
20 January 2021
10:00
Macarena Arenas
Abstract

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.

  • Junior Topology and Group Theory Seminar
2 December 2020
10:00
Davide Spriano
Abstract

 An important property of a Gromov hyperbolic space is that every path that is locally a quasi-geodesic is globally a quasi-geodesic. A theorem of Gromov states that this is a characterization of hyperbolicity, which means that all the properties of hyperbolic spaces and groups can be traced back to this simple fact. In this talk we generalize this property by considering only Morse quasi-geodesics.

We show that not only does this allow us to consider a much larger class of examples, such as CAT(0) spaces, hierarchically hyperbolic spaces and fundamental groups of 3-manifolds, but also we can effortlessly generalize several results from the theory of hyperbolic groups that were previously unknown in this generality.
 

  • Junior Topology and Group Theory Seminar
25 November 2020
10:00
Anna Parlak
Abstract

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. It is a generalisation of an older invariant, the Teichmüller polynomial, defined by McMullen in 2002.

The aim of my talk is to demonstrate that veering triangulations provide a convenient setup for computations. More precisely, I will use fairly easy arguments to obtain a fairly strong statement which generalises the results of McMullen relating the Teichmüller polynomial to the Alexander polynomial.

I will not assume any prior knowledge on the Alexander polynomial, the Teichmüller polynomial or veering triangulations.

The join button will be published on the right (Above the view all button) 30 minutes before the seminar starts (login required).

  • Junior Topology and Group Theory Seminar

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