Past Junior Topology and Group Theory Seminar

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.

The aim is introducing the Bieri-Neumann-Strebel invariants and showing some computations. These are geometric invariants of abstract groups that capture information about the finite generation of kernels of abelian quotients.

The Teichmüller space for a closed surface of genus g is the space of marked complex/hyperbolic structures on the surface. Teichmüller space also identifies with the space of Fuchsian representations of the fundamental group into PSL(2,R) (mod conjugation). Higher Teichmüller theory concerns special representations of surface (or hyperbolic) groups into higher rank Lie groups of non-compact type.

A well-known conjecture of Ivanov states that mapping class groups of surfaces with genus at least 3 virtually do not surject onto the integers. Putman and Wieland reformulated this conjecture in terms of higher Prym representations of finite-index subgroups of mapping class groups. We show that the Putman-Wieland conjecture holds for geometrically uniform subgroups. Along the way we construct a cover S of the genus 2 surface such that the lifts of simple closed curves do not generate the rational homology of S. This is joint work with Markovic.

We say that a graph G is Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. Examples include the Euclidean building of PSLn(Qp). We show that the rigidity of the building goes further by proving that a reconstruction is possible from only a partial local information, called “print”. We use this to prove the rigidity of graphs quasi-isometric to the building among which are the torsion-free lattices of PSLn(Qp).

Group theoretic Dehn filling, motivated by Dehn filling in the theory of 3- manifolds, is a process of constructing quotients of a given group. This technique is usually applied to groups with certain negative curvature feature, for example word-hyperbolic groups, to construct exotic and useful examples of groups. In this talk, I will start by recalling the notion of word-hyperbolic groups, and then show that how group theoretic Dehn filling can be used to answer the Burnside Problem and questions about mapping class groups of surfaces.

A big mapping class group is the mapping class group (MCG) of a surface of infinite type. Although several aspects of big MCGs remain mysterious, their geometric definition allows some simple, interesting arguments. In this talk, we will use big MCGs as an excuse to survey some (more or less) classical results in geometric group theory: we will present a quick introduction to infinite type surfaces, highlight differences between standard and large MCGs, and use Higman’s embedding theorem to deduce that there exists a big MCG that contains every finitely presented group as a subgroup.

Given a group G acting on a CAT(0) polygonal complex, X, it is natural to ask whether the structure of X allows us to deduce properties of G. We discuss some recent work on local properties that X may possess which allow us to answer these questions - in many cases we can in fact deduce that the group is a linear group over Z.

Orbifolds are a generalisation of manifolds which allow group actions to enter the picture. The most basic examples of orbifolds are quotients of manifolds by (non-free) finite group actions.
I will give an introduction to orbifolds, recalling a number of philosophically different but mathematically equivalent definitions. For starters, I will try to convince you that "a space locally modelled on a quotient of R^n by a finite group" is misleading. I will draw many pictures of orbifolds, make the connection to complexes of groups, and explain the definition of a map of orbifolds. In the process, I hope to demystify the definition of the orbifold fundamental group, the orbifold Euler characteristic and orbifold cohomology.

In this talk, I will discuss the important Grothendieck conjecture which originated Grothendieck-Teichmuller Theory, a bridge between Topology and Number Theory. On the geometric side, there is the study of automorphisms of mapping class groups that satisfy compatibility conditions with respect to subsurface inclusions. On the other side, there is the study of the absolute Galois group of the rationals, one of the most important objects in Number Theory today.
In my talk, I will introduce these objects and discuss the recent progress that has been made in understanding such automorphisms of mapping class groups. No background in Number Theory or Galois Theory is required.

Finiteness properties of groups provide various generalisations of the properties "finitely generated" and "finitely presented." We will define different types of finiteness properties and discuss Bestvina-Brady groups as they provide examples of groups with interesting combinations of finiteness properties.

Button observed that finitely generated linear groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated linear group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and some of its implications for the representation theory of certain 3-manifold groups.

Given an arbitrary group presentation, often very little can be deduced about the underlying group. It is thus something of a miracle that many properties of one-relator groups can be simply read-off from the defining relator. In this talk, I will discuss some of the classical results in the theory of one-relator groups, as well as the key trick used in many of their proofs. Time-permitting, I'll also discuss more recent work on this subject, including some open problems.

We introduce the notion of systolic complexes and give conditions on presentations to construct such complexes using Cayley graphs.

We consider Garside groups to find examples of groups admitting such a presentation.
 

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.