Past Junior Topology and Group Theory Seminar

A 3-manifold is a space which locally looks like R^3. A major theme in 3-manifold Topology is to understand and classify 3-manifolds. Given two compact 3-manifolds M_1,M_2 we can form another 3-manifold by taking what’s called the “connect sum” of M_1 and M_2. Under this operation, 3-manifolds can be decomposed uniquely into prime pieces just like the integers can be decomposed uniquely as a product of primes. We will discuss this prime decomposition theorem for 3-manifolds while also giving a wide variety of examples.

Dimension 4 is the first dimension in which exotic smooth manifold pairs appear — manifolds which are topologically the same but for which there is no smooth deformation of one into the other. On the other hand, smooth and PL manifolds (manifolds which can be described discretely) do coincide in dimension 4. Despite this, there has been comparatively little work done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. The aim of this talk will be to give a gentle introduction to some of the tools, techniques, and ideas, which inform a computational approach to 4-manifold topology.

The family of right-angled Artin groups (RAAGs) interpolates 
between free groups and free abelian groups. These groups are defined by 
a simplicial graph: the vertices correspond to generators, and two 
generators commute if and only if they are connected by an edge in the 
defining graph. A key feature of RAAGs is that many of their algebraic 
properties can be detected purely in terms of the combinatorics of the 
defining graph.
The family of outer automorphism groups of RAAGs similarly interpolates 
between Out(F_n) and GL(n, Z). While the l2-Betti numbers of GL(n, Z) 
are well understood, those of Out(F_n) remain largely mysterious. The 
aim of this talk is to introduce automorphism groups of RAAGs and to 
present a combinatorial criterion, expressed in terms of the defining 
graph, that characterizes when the first l2-Betti number of Out(RAAG) 
vanishes.
If time permits, we will also discuss higher l2-Betti numbers and 
algebraic fibring properties of these group

I will report on ongoing joint work with Sam Fisher on showing that the mapping class group has a finite index subgroup whose group ring embeds in a division ring. Our methods involve p-adic analytic groups, but no prior knowledge of this will be assumed and much of the talk will be devoted to explaining some of the underlying theory. Time permitting, I will also discuss some consequences for the profinite topology for the mapping class group and potential extensions to Out(RAAG).

A d-dimensional TQFT is a topological invariant which assigns (d-1)-dimensional manifolds to vector spaces and d-dimensional cobordisms to linear maps. In the early 90s, Reshetikhin and Turaev constructed examples of these in the case d=3, using the data of certain types of linear categories. In this talk, I will provide an overview of this construction, and then explore how this might be meaningfully extended downwards to assign 1-manifolds to "2-vector spaces". Minimal knowledge of category theory assumed!

You can cut a cake in half, a pizza into slices, but can you cut an infinite group? I'll tell you about my sensei's secret cutting technique and demonstrate a couple of examples. There might be some spectral goodies at the end too. 

Dehn surgery is a method of building three-dimensional manifolds that is ubiquitous throughout low-dimensional topology. I will give an introduction to Dehn surgery and discuss recent work with M. Kegel on the uniqueness of Dehn surgery descriptions of 3-manifolds. To do this, I will discuss the reason that Dehn surgery is so prominent - namely that it interacts very well with many structures, such as the geometry and gauge theory of 3-manifolds. (I will do my very best to assume very little background knowledge.)

Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups. 

Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that, in fact, it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.

In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

Recently, much attention has been paid to the intersection between coarse geometry and graph theory, giving rise to the fresh, exciting new field aptly known as ‘coarse graph theory’. One aspect of this area is the study of so-called ‘fat minors’, a large-scale analogue of the usual idea of a graph minor.

In this talk, I will introduce this area and motivate some interesting questions and conjectures. I will then sketch a proof that a finitely presented group is either virtually planar or contains arbitrarily ‘fat’ copies of every finite graph.

No prior knowledge or passion for graph theory will be assumed in this talk.

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.

There is a well-known class of knots, called torus knots, which are those that can be drawn on a "standardly embedded" torus (one that separates the 3-sphere into two solid tori). A fairly natural property of other knots to consider is the genus necessary for that knot to be drawn on a standardly embedded genus g surface. This knot invariant has been studied under the name "embeddability". The goal of this talk is to introduce the invariant, look at some upper and lower bounds in terms of other invariants, and examine its behavior under connected sum.

Property (T) is a rigidity property for group representations. It is generally very difficult to determine whether an infinite group has property (T) or not. It has long been known that a discrete group with a finite symmetric generating set has property (T) if and only if the group Laplacian is a positive element in the maximal group C*-algebra. However, this characterization has not been useful in addressing the question for automorphism groups of (non-abelian) free groups. In his 2016 paper, Ozawa proved that the phenomenon of 'positivity' of the group Laplacian is observed in the real group algebra, meaning that the Laplacian can be decomposed into a 'sum of squares'. This result transformed checking property (T) into a finite-dimensional condition that can be performed with the assistance of computers. In this talk, we will introduce property (T) and discuss Ozawa's result in detail.

The BNSR Invariant is a classical geometric invariant that encodes the finite generation of all coabelian subgroups of a given finitely generated group. The aim of this talk is to present a conjecture about the structure of the BNSR invariant of an Artin group and to present a new family in which the conjecture is true in terms of graph colorings.

We introduce ultrasolid modules, a variant of complete topological vector spaces. In this setting, we will prove some results in commutative algebra and apply them to the deformation of algebraic varieties in the language of derived algebraic geometry.

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.

The hyperbolic plane and its higher-dimensional analogues are well-known
objects. They belong to a larger class of spaces, called rank-one
symmetric spaces, which include not only the hyperbolic spaces but also
their complex and quaternionic counterparts, and the octonionic
hyperbolic plane. By a result of Pansu, two of these families exhibit
strong rigidity properties with respect to their self-quasiisometries:
any self-quasiisometry of a quaternionic hyperbolic space or the
octonionic hyperbolic plane is at uniformly bounded distance from an
isometry. The goal of this talk is to give an overview of the rank-one
symmetric spaces and the tools used to prove Pansu's rigidity theorem,
such as the subRiemannian structure of their visual boundaries and the
analysis of quasiconformal maps.

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.

Donaldson proved that there are pairs of 4-manifolds that are homeomorphic but not diffeomorphic, a phenomenon that does not appear for any lower dimensional manifolds. Until recently, proving this for compact manifolds has required smooth 4-manifold invariants coming from gauge theory. In this talk, we will give an introduction to an exciting new smooth 4-manifold invariant of Morrison Walker and Wedich, called a skein lasagna module that does not rely on gauge theory. Further, this talk will not assume any knowledge of 4-manifold topology.

In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

In 1911, Otto Toeplitz posed the intriguing "Square Peg Problem," asking whether every Jordan curve admits an inscribed square. Despite over a century of study, the problem remains unsolved in its full generality. However, significant progress has been made over the years. In this talk, we explore recent advancements by Andrew Lobb and Joshua Greene, who approach the problem through the lens of Lagrangian Floer homology. Specifically, we outline a proof of their result: every smooth Jordan curve inscribes every rectangle up to similarity.

Embedding the group algebra into a division ring has proven to be a powerful tool for detecting structural properties of the group, especially in relation to its homology. In this talk, we will show how division rings can be used to identify residual properties of groups, one-ended groups, and coherent groups. We will place special emphasis on the class of free-by-cyclic groups to provide a clear, explicit exposition.

The McCullough-Miller space is a contractible simplicial complex that admits an action of the pure symmetric (outer) automorphisms of the free group, with stabilizers that are free abelian. This space has been used to derive several cohomological properties of these groups, such as computing their cohomology ring and proving that they are duality groups. In this talk, we will generalize the construction to right-angled Artin groups (RAAGs), and use it to obtain some interesting cohomological results about the pure symmetric (outer) automorphisms of RAAGs.

A topological quantum field theory (TQFT) is a functor from a category of bordisms to a category of vector spaces. Classifying low-dimensional TQFTs often involves considering presentations of bordism categories in terms of generators and relations. In this talk, we will introduce these concepts and outline a program for obtaining such presentations using Morse–Cerf theory.

The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.

A group is coherent if all its finitely generated subgroups are finitely presented. Aside from some easy cases, it appears that coherence is a phenomenon that occurs only among groups of cohomological dimension 2. In this talk, we will give many examples of coherent and incoherent groups, discuss techniques to prove a group is coherent, and mention some open problems in the area.

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

If \(F\) is a free group and \(F/N\) is a presentation of a group \(G\), there is a natural way to turn the abelianisation of \(N\) into a \(\mathbb ZG\)-module, known as the relation module of the presentation. The images of normal generators for \(N\) yield \(\mathbb ZG\)-module generators of the relation module, but 'lifting' \(\mathbb ZG\)-generators to normal generators cannot always be done by a result of Dunwoody. Nevertheless, it is an open problem, known as the relation gap problem, whether the relation module can have strictly fewer \(\mathbb ZG\)-module generators than \(N\) can have normal generators when \(G\) is finitely presented. In this talk I will survey what is known and what is not known about this problem and its variations and discuss some recent progress for groups with a cyclic relation module.

In 2006, Jan Dymara introduced the concept of weighted \(\ell^2\) Betti numbers as a method of computing regular \(\ell^2\) Betti numbers of buildings. This notion of dimension is measured by using Hecke algebras associated to the relevant Coxeter groups. I will briefly introduce buildings and then give a comparison between the regular \(\ell^2\) Betti numbers and the weighted ones.

This talk will introduce Khovanov and Knot Floer Homology as tools for studying knots. I will then cover some applications to problems in knot theory including distinguishing embedded surfaces and how they can be used in the context of ribbon concordances. No prior knowledge of either will be necessary and lots of pictures are included.

We will try to solve the isomorphism problem amongst Coxeter groups by looking at finite quotients.  Some success is found in the classes of affine and right-angled Coxeter groups.  Based on joint work with Samuel Corson, Philip Moeller, and Olga Varghese.

This talk will serve as an introduction to the outer automorphism group of a free group, its properties and the objects used to study it: especially train track maps (with various adjectives) and Culler--Vogtmann outer space. If time allows I will discuss recent work joint with Hillen, Lyman and Pfaff on stretch factors in rank 3, but the goal of the talk will be to introduce the topic well rather than to speedrun towards the theorem.

I'll talk about the Morse local-to-global property and try to convince you that is a good property. There are three reasons. Firstly, it is satisfied by many examples of interest. Secondly, it allows to prove many theorems. Thirdly, it sits nicely in the larger program of classifying groups up to quasi-isometry and it has connections with open questions.

RFRS groups were introduced by Ian Agol in connection with virtual fibering of 3-manifolds. Notably, the class of RFRS groups contains all compact special groups, which are groups with particularly nice cocompact actions on cube complexes. In this talk, I will give an introduction to ℓ²-Betti numbers from an algebraic perspective and discuss what group theoretic properties we can conclude from the (non)vanishing of the ℓ²-Betti numbers of a RFRS group.

It is a standard result that mapping class groups of high genus do not surject the integers. This is easily shown by computing the abelianization of the mapping class group using a presentation. Once we pass to finite index subgroups, this becomes a conjecture of Ivanov. More generally, we can ask which groups admit epimorphisms from finite index subgroups of the mapping class group. In this talk, I will present a geometric approach to this question, using harmonic maps, and explain some recent results.

We will give a relaxed introduction to some of the most classical dynamical systems - Anosov flows. These flows were highly influential in the development of ideas which the audience might be more familiar with. For example, Anosov flows give rise to exponential group growth and taut foliations, both of which we will discuss. Finally, we will talk about some recent work obstructing Anosov flows and their combinatorial analogs - veering triangulations

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).

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.

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.

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.