Past Junior Topology and Group Theory Seminar

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.

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.

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.

A fundamental result in 3-manifold topology is the loop theorem: Given a null-homotopic simple closed curve in the boundary of a compact 3-manifold $M$, it bounds an embedded disk in $M$. The standard topological proof of this uses the tower construction due to Papakyriakopoulos. In this talk, I will introduce basic existence and regularity results on minimal surfaces, and show how to use the tower construction to prove a geometric version of the loop theorem due to Meeks--Yau: Given a null-homotopic simple closed curve in the boundary of a compact Riemannian 3-manifold $M$ with convex boundary, it bounds an embedded disk of least area. This also gives an independent proof of the (topological) loop theorem.

Many classes of finitely generated groups have been studied using formal language theory techniques. One historical example is the study of geodesics, which gives rise to the strict growth series of a group. Properties of languages associated to groups can provide insight into the nature of the growth series.

In this talk we will introduce languages associated to conjugacy classes, rather than elements of the group. This will lead us to define an analogous series, namely the conjugacy growth series of a group, which has become a popular topic in recent years. After discussing the necessary group theoretic and language tools needed, we will focus on how these conjugacy languages behave in graph products. We will finish with some new results which look at when these properties can extend to virtual graph products.

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

The handlebody group is defined as the mapping class group of a three-dimensional handlebody. We will survey some geometric and algebraic properties of the handlebody groups and compare them to those of two of the most studied (classes of) groups in geometric group theory, namely mapping class groups of surfaces, and ${\rm Out}(F_n)$. We will also introduce the disk graph, the handlebody-analogon of the curve graph of a surface, and discuss some of its properties.

Given a group of type ${\rm FP}_n$, one may ask if this property also holds for its subgroups. The BNS invariant is a subset of the character sphere that fully captures this information for subgroups that are kernels of characters. It also provides an interesting connection of finiteness properties of subgroups and group homology. In this talk I am going to give an introduction to this problem and present an attempt to generalize the BNS invariant to more subgroups than just the kernels of characters.

In recent years, a tend of higher category theory is growing from multiple areas of research throughout mathematics, physics and theoretical computer science. Guided by Cobordism Hypothesis, I would like to introduce some basics of `higher representation theory’, i.e. the part of higher category theory where we focus on the fundamental objects: `finite dimensional’ linear n-categories. If time permits, I will also introduce some recent progress in linear higher categories and motivations from condensed matter physics.

Persistent homology is both a powerful framework for data science and a fruitful source of mathematical questions. Here, we will give an introduction to both single-parameter and multiparameter persistent homology. We will see some examples of how topology has been successfully applied to the real world, and also explore some of the pure-mathematical ideas that arise from this new perspective.

Separability is an algebraic property enjoyed by certain subsets of groups. In the world of non-positively curved groups, it has a not-too-well-understood link to geometric properties such as convexity. We explore this connection in the setting of relatively hyperbolic groups and discuss a recent joint work in this area involving products of quasiconvex subgroups.

We review a few instances in which the first $\ell^2$ Betti number of a group is a profinite invariant and we discuss some applications and open problems.

We survey the theory of $\ell^2$-invariants, their applications in group theory and topology, and introduce a positive characteristic version of $\ell^2$-theory. We also discuss the Atiyah and Lück approximation conjectures, two of the central problems in this area.

Profinite rigidity is essentially the study of which groups can be distinguished from each other by their finite quotients. This talk is meant to give a gentle introduction to the area - I will explain which questions are the right ones to ask and give an overview of some of the main results in the field. I will assume knowledge of what a group presentation is.

After a general introduction to the study of random walks on groups, we discuss the relationship between limit theorems for random walks on Lie groups and Diophantine properties of the underlying distribution. Indeed, we will discuss the classical abelian case and more recent results by Bourgain-Gamburd for compact simple Lie groups such as SO(3). If time permits, we discuss some new results for non-compact simple Lie groups such as SL_2(R). We hope to touch on the relevant methods from harmonic analysis, number theory and additive combinatorics. The talk is aimed at a general audience. 

A branched covering between two surfaces looks like a regular covering map except for finitely many branching points, where some non-trivial ramification may occur. Informally speaking, the existence problem asks whether we can find a branched covering with prescribed behaviour around its branching points.

A variety of techniques have historically been employed to tackle this problem, ranging from studying representations of surface groups into symmetric groups to drawing "dessins d'enfant" on the covering surface. After introducing these techniques and explaining how they can be applied to the existence problem, I will briefly present a conjecture unexpectedly relating branched coverings and prime numbers.
 

This will be a gentle introduction to the theory of pseudo-Anosov  flows on 3-manifolds, as seen from the perspective of a topologist and not a dynamicist.

I will start by considering geodesic flows on the unit tangent bundles of hyperbolic surfaces. This will lead to a definition of an Anosov and then a pseudo-Anosov flow on a 3-manifold. After discussing a couple of examples, I will outline some connections between pseudo-Anosov flows and other aspects of 3-manifold topology/ geometry/ group theory.

We will give a fairly self contained introduction to acylindrically hyperbolic groups, using mapping class groups as a motivating example. This will be a mainly expository talk, the aim is to make my topology seminar talk in week 5 more accessible to people who are less familiar with these topics.

The question of when you can quasiisometrically embed a solvable Baumslag-Solitar group into another turns out to be equivalent to the question of when you can (1,A)-quasiisometrically embed a rooted tree into another rooted tree. We will briefly describe the geometry of the solvable Baumslag-Solitar groups before attacking the problem of embedding trees. We will find that the existence of (1,A)-quasiisometric embeddings between trees is intimately related to the boundedness of a family of integer sequences. 

Knot projections for knots in the 3-sphere allow us to easily describe knots, compute invariants, enumerate all knots, manipulate them under Reidemister moves and feed them into a computer. One might hope for a similar representation of knots in general 3-manifolds. We will survey properties of knots in general 3-manifolds and discuss a proposed diagram-esque representation of them.

Amenable actions are answering the question: "When can we prevent things like the Banach-Tarski Paradox happening?". It turns out that the most intuitive measure-theoretic sufficient condition is also necessary. We will briefly discuss the paradox, prove the equivalent conditions for amenability, give some ways of producing interesting examples of amenable groups and talk about amenable groups which can't be produced in these 'elementary' ways.

Teaser question: show that you can't decompose Z into finitely many pieces, which after rearrangement by translations make two copies of Z. (I.e. that you can't get the Banach-Tarski paradox on Z.)

The Gromov boundary of a hyperbolic group is a useful topological invariant, the properties of which can encode all sorts of algebraic information. It has found application to some algorithmic questions, such as finding finite splittings (Dahmani-Groves) and, more recently, computing JSJ-decompositions (Barrett). In this talk we will introduce the boundary of a hyperbolic group. We'll outline how one can approximate the boundary with "large spheres" in the Cayley graph, in order to search for topological features. Finally, we will also discuss how this idea is applied in the aforementioned results. 

We will record some surprising and lesser-known properties of free groups, and use these to give a model theoretic analysis of free group automorphisms and orbits under Aut(F). This will result in a neat geometric description of (a logic-flavoured analogue of) algebraic closures in a free group. An almost immediate corollary will be that elementary subgroups of a free group are free factors.

I will assume no familiarity with first-order logic and model theory - the beginning of the talk will be devoted to familiarize everyone with the few required notions.