Past Junior Topology and Group Theory Seminar

Deciding whether or not two elements of a group are conjugate might seem like a trivial problem. However, there exist finitely presented groups where this problem is undecidable: there is no algorithm to output yes or no for any two elements chosen. In this talk Houghton groups (a family of groups all having solvable conjugacy problem) will be introduced as will the idea of twisted conjugacy: a generalisation of the conjugacy problem where an automorphism is also given. This will be our main tool in answering whether finite extensions and finite index subgroups of any Houghton group have solvable conjugacy problem.

We saw earlier that a subquadratic isoperimetric inequality implies a linear one. I will give examples of groups, due to Brady and Bridson, which prove that this is the only gap in the isoperimetric spectrum. 

In 1983 Kerckhoff settled a long standing conjecture by Nielsen proving that every finite subgroup of the mapping class group of a compact surface can be realized as a group of diffeomorphisms. An important consequence of this theorem is that one can now try to study subgroups of the mapping class group taking the quotient of the surface by these groups of diffeomorphisms. In this talk we will study quotients of surfaces under the action of a finite group to find bounds on the cardinality of such a group.

The Dehn function of a group measures the complexity of the group's word problem, being the upper bound on the number of relations from a group presentation required to prove that a word in the generators represents the identity element. The Filling Theorem which was first stated by Gromov connects this to the isoperimetric functions of Riemannian manifolds. In this talk, we will see the classification of hyperbolic groups as those with a linear Dehn function, and give Bowditch's proof that a subquadratic isoperimetric inequality implies a linear one (which gives the only gap in the "isoperimetric spectrum" of exponents of polynomial Dehn functions).

We will give an outline of the proof by Kahn and Markovic who showed that a closed hyperbolic 3-manifold $\textbf{M}$ contains a closed $\pi_1$-injective surface. This is done using exponential mixing to find many pairs of pants in $\textbf{M}$, which can then be glued together to form a suitable surface. This answers a long standing conjecture of Waldhausen and is a key ingredient in the proof of the Virtual Haken Theorem.

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this talk we will discuss a similar structure theorem for minimal vertex cuts separating the ends of a graph; these can be encoded by a succulent, a mild generalization of a cactus that is still tree-like.

After motivating why we would like to find examples of simple totally disconnected locally compact groups, I will describe a construction due to Banks, Elder and Willis which yields infinitely many such examples when given certain groups acting on a tree.

I shall outline a general method for finding upper bounds on the diameters of finite groups, based on the Solovay-Kitaev procedure from quantum computation. This method may be fruitfully applied to groups arising as quotients of many familiar pro-p groups. Time permitting, I will indicate a connection with weak spectral gap, and give some applications.

Waldhausen defined higher K-groups for categories with certain extra structure. In this talk I will define categories with cofibrations and weak equivalences, outline Waldhausen's construction of the associated K-Theory space, mention a few important theorems and give some examples. If time permits I will discuss the infinite loop space structure on the K-Theory space.

Subgroup separability is a group-theoretic property that has important implications for geometry and topology, because it allows us to lift immersions to embeddings in a finite sheeted covering space. I will describe how this works in the case of graphs, and go on to motivate the construction of special cube complexes as an attempt to generalise the technique to higher dimensions.

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

Brady showed that there are hyperbolic groups with non-hyperbolic finitely presented subgroups. I will present a new construction of this kind using Bestvina-Brady Morse theory.

In some of their recent work Derbez and Wang studied volumes of representations of 3-manifold groups into the Lie groups $$Iso_e \widetilde{SL_2(\mathbb{R})} \mbox{ and }PSL(2,\mathbb{C}).$$ They computed the set of all volumes of representations for a fixed prime closed oriented 3-manifold with $$\widetilde{SL_2(\mathbb{R})}\mbox{-geometry}$$ and used this result to compute some volumes of Graph manifolds after passing to finite coverings.

In the talk I will give a brief introduction to the theory of volumes of representations and state some of Derbez' and Wang's results. Then I will prove an additivity formula for volumes of representations into $$Iso_e \widetilde{SL_2(\mathbb{R})}$$ which enables us to improve some of the results of Derbez and Wang.

I will explain why one can symplectically embed closed symplectic manifolds (with integral symplectic form) into CPⁿ and compute the weak homotopy type of the space of all symplectic embeddings of such a symplectic manifold into CP^∞.

The notion of automatic groups emerged from conversations between Bill Thurston and Jim Cannon on the nice algorithmic properties of Kleinian groups. In this introductory talk we will define automatic groups and then discuss why they are interesting. This centres on how automatic groups subsume many other classes of groups (e.g. hyperbolic groups, finitely generated Coxeter groups, and braid groups) and have good properties (e.g. finite presentability, fast solution to the word problem, and type FP_∞).

Many interesting properties of groups are inherited by their subgroups examples of such are finiteness, residual finiteness and being free. People have asked whether hyperbolicity is inherited by subgroups, there are a few counterexamples in this area. I will be detailing the proof of some of these including a construction due to Rips of a finitely generated not finitely presented subgroup of a hyperbolic group and an example of a finitely presented subgroup which is not hyperbolic.

I will introduce and motivate the concept of largeness of a group. I will then show how tools from different areas of mathematics can be applied to show that all free-by-cyclic groups are large (and try to convince you that this is a good thing).

I will show how to construct an infinite family of totally geodesic surfaces in the figure eight knot complement that do not remain totally geodesic under certain Dehn surgeries. If time permits, I will explain how this behaviour can be understood via the theory of quadratic forms.

We will discuss TQFTs (at a basic level), then higher categorical extensions, and see how these lead naturally to the notion of Segal spaces.

Last week in the Kinderseminar I talked about a rough estimate on volumes of certain hyperbolic 3-manifolds. This time I will describe a different approach for similar estimates (you will not need to remember that talk, don't worry!), which is, in some sense, complementary to that one, as it regards mapping tori. A theorem of Jeffrey Brock provides bounds for their volume in terms of how the monodromy map acts on the pants graph (a relative of the better known curve complex) of the base surface. I will describe the setting and the relevance of this result (in particular the one it has for me); hopefully, I will also tell you part of its proof.

A bit more than ten years ago, Peter Oszváth and Zoltán Szabó defined Heegaard-Floer homology, a gauge theory inspired invariant of three-manifolds that is designed to be more computable than its cousins, the Donaldson and Seiberg-Witten invariants for four-manifolds. This invariant is defined in terms of a Heegaard splitting of the three-manifold. In this talk I will show how Heegaard-Floer homology is defined (modulo the analysis that goes into it) and explain some of the directions in which people have taken this theory, such as knot theory and fitting Heegaard-Floer homology into the scheme of topological field theories.