Forthcoming events in this series


Wed, 06 Mar 2024

16:00 - 17:00
L6

TBA

Michael Schmalian
(University of Oxford)
Wed, 28 Feb 2024

16:00 - 17:00
L6

Revisiting property (T)

Ismael Morales
(University of Oxford)
Abstract

Property (T) was introduced by Kazhdan in the sixties to show that lattices in higher rank semisimple Lie groups are finitely generated. We will discuss some classical examples of groups that satisfy this property, with a particular focus on SL(3, R).

Wed, 21 Feb 2024
16:00
L6

Groups Acting Acylindrically on Trees

William Cohen
(University of Cambridge)
Abstract

It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

Wed, 14 Feb 2024

16:00 - 17:00
L6

One-ended graph braid groups and where to find them

Ruta Sliazkaite
(University of Warwick)
Abstract

Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.

Wed, 07 Feb 2024

16:00 - 17:00
L6

Relationships between hyperbolic and classic knot invatiants

Colin McCulloch
(University of Oxford)
Abstract

For a hyperbolic knot there are two types of invariants, the hyperbolic invariants coming from the geometric structure and the classical invariants coming from the topology or combinatorics. It has been observed in many different cases that these seemingly different types of invariants are in fact related. I will give examples of these relationships and discuss in particular a link by Stoimenow between the determinant and volume.  

Wed, 31 Jan 2024

16:00 - 17:00
L6

Distinguishing free-by-(finite cyclic) groups by their finite quotients

Paweł Piwek
(University of Oxford)
Abstract
Finitely generated free-by-(finite cyclic) groups turn out to be distinguished from each other by their finite quotients - and this is thanks to being very constrained by their finite subgroups and their centralisers. This has a consequence to distinguishing in the same way the free-by-cyclic groups with centre. This is joint work with Martin Bridson.
Wed, 17 Jan 2024

16:00 - 17:00
L6

Spectra of surfaces and MCG actions on random covers

Adam Klukowski
(University of Oxford)
Abstract

The Ivanov conjecture is equivalent to the statement that every covering map of surfaces has the so-called Putman-Wieland property. I will discuss my recent work with Vlad Marković, where we prove it for asymptotically all coverings as the degree grows. I will give some overview of our main tool: spectral geometry, which is related to objects like the heat kernel of a hyperbolic surface, or Cheeger connectivity constant.

Wed, 29 Nov 2023

16:00 - 17:00
L6

Combinatorial Hierarchical Hyperbolicity of the Mapping Class Group

Kaitlin Ragosta
(Brandeis University)
Abstract

The mapping class group of a surface has a hierarchical structure in which the geometry of the group can be seen by examining its action on the curve graph of every subsurface. This behavior was one of the motivating examples for a generalization of hyperbolicity called hierarchical hyperbolicity. Hierarchical hyperbolicity has many desirable consequences, but the definition is long, and proving that a group satisfies it is generally difficult. This difficulty motivated the introduction of a new condition called combinatorial hierarchical hyperbolicity by Behrstock, Hagen, Martin, and Sisto in 2020 which implies the original and is more straightforward to check. In recent work, Hagen, Mangioni, and Sisto developed a method for building a combinatorial hierarchically hyperbolic structure from a (sufficiently nice) hierarchically hyperbolic one. The goal of this talk is to describe their construction in the case of the mapping class group and illustrate some of the parallels between the combinatorial structure and the original. 

Wed, 22 Nov 2023

16:00 - 17:00
L6

3-manifold algorithms, representation theory, and the generalised Riemann hypothesis

Adele Jackson
(University of Oxford)
Abstract

You may be surprised to see the generalised Riemann hypothesis appear in algorithmic topology. For example, knottedness was originally shown to be in NP under the assumption of GRH.
Where does this condition come from? We will discuss this in the context of 3-sphere recognition, and examine why the approach fails for higher dimensions.

Wed, 15 Nov 2023

16:00 - 17:00
L6

Fáry-Milnor type theorems

Shaked Bader
(University of Oxford)
Abstract
In 1947 Karol Borsuk conjectured that if an ant is walking on a circle embedded piecewise linearly in 3 and is not dizzy (did not wind around itself twice) then the circle bounds a disc. He actually phrased it as follows: the total curvature of a knotted knot must be at least 4π
One may ask the same question with other spaces instead of 3.
We will present Milnor's proof of the classical conjecture, then define CAT(0) spaces and present some ideas from Stadler's proof in that setting and a more elementary proof in the setting of CAT(0) polygonal complexes.
 
Wed, 08 Nov 2023

16:00 - 17:00
L6

Navigating the curve graph with train tracks

Filippo Baroni
(University of Oxford)
Abstract

It is a truth universally acknowledged, that an infinite group in possession of a good algebraic structure, must be in want of a hyperbolic space to act on. For the mapping class group of a surface, one of the most popular choices is the curve graph. This is a combinatorial object, built from curves on the surface and intersection patterns between them.
Hyperbolicity of the curve graph was proved by Masur and Minsky in a celebrated paper in 1999. In the same article, they showed how the geometry of the action on this graph reflects dynamical/topological properties of the mapping class group; in particular, loxodromic elements are precisely the pseudo-Anosov mapping classes.
In light of this, one would like to better understand distances in the curve graph. The graph is locally infinite, and finding a shortest path between two vertices is highly non-trivial. In this talk, we will see how to use the machinery of train tracks to overcome this issue and compute (approximate) distances in the curve graph. If time permits -- which, somehow, it never does -- we will also analyse this construction from an algorithmic perspective.

Wed, 01 Nov 2023

16:00 - 17:00
L6

Topology and dynamics on the space of subgroups

Pénélope Azuelos
(University of Bristol)
Abstract

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.

Wed, 25 Oct 2023

16:00 - 17:00
L6

Alternating knots and branched double covers

Soheil Azarpendar
(University of Oxford)
Abstract

An old and challenging conjecture proposed by R.H. Fox in 1962 states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot are trapezoidal i.e. strictly increase, possibly plateau, then strictly decrease. We give a survey of the known results and use them to motivate the study of branched double covers. The second part of the talk focuses on the properties of the branched double covers of alternating knots.

Wed, 18 Oct 2023

16:00 - 17:00
L6

Fibring in manifolds and groups

Monika Kudlinska
(University of Oxford)
Abstract

Algebraic fibring is the group-theoretic analogue of fibration over the circle for manifolds. Generalising the work of Agol on hyperbolic 3-manifolds, Kielak showed that many groups virtually fibre. In this talk we will discuss the geometry of groups which fibre, with some fun applications to Poincare duality groups - groups whose homology and cohomology invariants satisfy a Poincare-Lefschetz type duality, like those of manifolds - as well as to exotic subgroups of Gromov hyperbolic groups. No prior knowledge of these topics will be assumed.

Disclaimer: This talk will contain many manifolds.

Wed, 11 Oct 2023
16:00
L6

Reasons to be accessible

Joseph MacManus
(University of Oxford)
Abstract

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

Wed, 14 Jun 2023
16:00
L6

Asymptotic dimension of groups

Panagiotis Tselekidis
(University of Oxford)
Abstract

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

Wed, 07 Jun 2023
16:00
L6

TBC

TBC
Wed, 31 May 2023
16:00
L6

Accessibility, QI-rigidity, and planar graphs

Joseph MacManus
(University of Oxford)
Abstract

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.

Wed, 17 May 2023
16:00
L6

A brief history of virtual Haken

Filippo Baroni
(University of Oxford)
Abstract

The virtual Haken theorem is one of the most influential recent results in 3-manifold theory. The statement dates back to Waldhausen, who conjectured that every aspherical closed 3-manifold has a finite cover containing an essential embedded closed surface. The proof is usually attributed to Agol, although his virtual special theorem is only the last piece of the puzzle. This talk is dedicated to the unsung heroes of virtual Haken, the mathematicians whose invaluable work helped turning this conjecture into a theorem. We will trace the history of a mathematical thread that connects Thurston-Perelman's geometrisation to Agol's final contribution, surveying Kahn-Markovic's surface subgroup theorem, Bergeron-Wise's cubulation of 3-manifold groups, Haglund-Wise's special cube complexes, Wise's work on quasi-convex hierarchies and Agol-Groves-Manning's weak separation theorem.

Wed, 10 May 2023
16:00
L6

Vanishing of group cohomology, Kazhdan’s Property (T), and computer proofs

Piotr Mizerka
(Polish Academy of Sciences)
Abstract

We will look at the vanishing of group cohomology from the perspective of Kazhdan’s property (T). We will investigate an analogue of this property for any degree, introduced by U. Bader and P. W. Nowak in 2020 and describe a method of proving these properties with computers.

Wed, 03 May 2023
16:00
L6

A Motivation for Studying Hyperbolic Cusps

Misha Schmalian
(University of Oxford)
Abstract

We will give an introduction to hyperbolic cusps and their Dehn fillings. In particular, we will give a brief survey of quantitive results in the field. To motivate this work, we will sketch how these techniques are used for studying the classical question of characteristic slopes on knots.

Wed, 26 Apr 2023
16:00
L6

Insufficiency of simple closed curve homology

Adam Klukowski
(University of Oxford)
Abstract

This talk is concerned with the question of generating the homology of a covering space by lifts of simple closed curves (from topological viewpoint), and generating the first homology of a subgroup by powers of elements outside certain filtrations (from group-theoretic viewpoint). I will sketch Malestein's and Putman's construction of examples of branched covers where lifts of scc's span a proper subspace. I will discuss the relation of their proof to the Magnus embedding, and present recent results on similar embeddings of surface groups which facilitate extending their theorems to unbranched covers.

Wed, 08 Mar 2023
16:00
L6

99 problems and presentations are most of them

Naomi Andrew
(University of Oxford)
Abstract

Geometric (even combinatorial) group theory suffers from the unfortunate situation that many obvious questions about group presentations (ex: is this a presentation of the trivial group? is this word the identity in that group?) cannot be answered. Not only "we don't know how to tell" but "we know that we cannot know how to tell" - this is called undecidability. This talk will serve as an introduction (for non-experts, since I am also such) to the area of group theoretic decision problems: I'll aim to cover some problems, some solutions (or half-solutions) and some of the general sources of undecidability, as well as featuring some of my (least?) favourite pathological groups.

Wed, 01 Mar 2023
16:00
L6

Algorithms and 3-manifolds

Adele Jackson
(University of Oxford)
Abstract

Given a mathematical object, what can you compute about it? In some settings, you cannot say very much. Given an arbitrary group presentation, for example, there is no procedure to decide whether the group is trivial. In 3-manifolds, however, algorithms are a fruitful and active area of study (and some of them are even implementable!). One of the main tools in this area is normal surface theory, which allows us to describe interesting surfaces in a 3-manifold with respect to an arbitrary triangulation. I will discuss some results in this area, particularly around Seifert fibered spaces.