Past Junior Topology and Group Theory Seminar

This talk will be an introduction to the weird and wonderful world of Thompson's groups $F$, $T$ and $V$. For example, the group $T$ was the first known finitely presented infinite simple group, $V$ has a finitely presented subgroup with co-NP-complete word problem, and whether or not $F$ is amenable is an infamous open problem.

Localization and completion of spaces are fundamental tools in homotopy theory. "Zabrodsky mixing" uses localization to "mix homotopy types". It was used to provide a counterexample to the conjecture that any finite H-space which is $A_3$ is also $A_\infty$. The material in this talk will be very classical (and rather basic). I will describe Sullivan's localization functor and demonstrate Zabrodsky's mixing by constructing a non-classical H-space.

A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)

A Kähler group is a group which is isomorphic to the fundamental group of a compact Kähler manifold. In 2008 Dimca and Suciu proved that the groups which are both Kähler and isomorphic to the fundamental group of a closed 3-manifold are precisely the finite subgroups of $O(4)$ which act freely on $S^3$. In this talk we will explain Kotschick's proof of this result. On the 3-manifold side the main tools that will be used are the first Betti number and Poincare Duality and on the Kähler group side we will make use of the Albanese map and some basic results about Kähler groups. All relevant notions will be explained in the talk.

In Bass-Serre theory, one derives structural properties of groups from their actions on simplicial trees. In this talk, we introduce the theory of groups acting on $\mathbb{R}$-trees. In particular, we explain how the Rips machine is used to classify finitely generated groups which act freely on $\mathbb{R}$-trees.

I will look at some decidability questions for subgroups of Aut($F_n$) for general $n$. I will then discuss semisimple actions of Aut($F_n$) on complete CAT(0) spaces proving that the Nielsen moves will act elliptically. I will also look at proving Aut($F_3$) is large and if time permits discuss the fact that Aut($F_n$) is not Kähler

This talk will give an almost complete proof of the h-cobordism theorem, paying special attention to the sources of the dimensional restrictions in the theorem. If time allows, the alterations needed to prove its cousin, the s-cobordism theorem, will also be sketched.

The Nottingham Group of a finite field is an object of great interest in profinite group theory, owing to its extreme structural properties and the relative ease with which explicit computations can be made within it. In this talk I shall explore both of these themes, before describing some new work on efficient short-word approximation in the Nottingham Group, based on the profinite Solovay-Kitaev procedure. Time permitting, I shall give an application to the dynamics of compositions of random power series.

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^∞.