I will discuss the basics of normal surface theory, and how they were used to give an algorithm for deciding whether a given diagram represents the unknot. This version is primarily based on Haken's work, with simplifications from Schubert and Jaco-Oertel.

# Past Junior Topology and Group Theory Seminar

Polycyclic groups either have polynomial growth, in which case they are virtually nilpotent, or exponential growth. I will give two interesting examples of "small" polycyclic groups which are extensions of $\mathbb{R}^2$ and the Heisenberg group by the integers, and attempt to justify the claim that they are small by sketching an argument that every exponential growth polycyclic group contains one of these.

You’re an amateur investigator hired to uncover the mysterious goings on of a dark cult. They call themselves Geometric Group Theorists and they’re under suspicion of pushing humanity’s knowledge too far. You’ve tracked them down to their supposed headquarters. Foolishly, you enter. Your mind writhes as you gaze unwittingly upon the Eldritch horror they’ve summoned… Group Theory! You think fast; donning the foggy glasses of quasi-isometry, you prevent your mind shattering from the unfathomable complexity of The Beast. You spy a weak spot and the phrase `Gromov Hyperbolicity’ flashes across your mind. You peer deeper, further, forever… only to find yourself somewhere rather familiar, strange, but familiar… no, self-similar! You’ve fought with fractals before, this weirdness can be tamed! Your insight is sufficient and The Beast retreats for now.

In other words, given an infinite group, we associate to it an infinite graph, called a Cayley graph, which gives us a notion of the ‘geometry’ of a group. Through this we can ask what kind of groups have hyperbolic geometry, or at least an approximation of it called Gromov hyperbolicity. Hyperbolic groups are quite a nice class of groups but a large one, so we introduce the Gromov boundary of a hyperbolic group and explain how it can be used to distinguish groups in this class.

A core problem in the study of manifolds and their topology is that of telling them apart. That is, when can we say whether or not two manifolds are homeomorphic? In two dimensions, the situation is simple, the Classification Theorem for Surfaces allows us to differentiate between any two closed surfaces. In three dimensions, the problem is a lot harder, as the century long search for a proof of the Poincaré Conjecture demonstrates, and is still an active area of study today.

As an early pioneer in the area of 3-manifolds Seifert carved out his own corner of the landscape instead of attempting to tackle the entire problem. By reducing his scope to the subclass of 3-manifolds which are today known as Seifert fibred spaces, Seifert was able to use our knowledge of 2-manifolds and produce a classification theorem of his own.

In this talk I will define Seifert fibred spaces, explain what makes them so much easier to understand than the rest of the pack, and give some insight on why we still care about them today.

"Fibre theorems" in the style of Quillen's fibre lemma are versatile tools used to study the topology of partially ordered sets. In this talk, I will formulate two of them and explain how these can be used to determine the homotopy type of the complex of (conjugacy classes of) free factors of a free group.

The latter is joint work with Radhika Gupta (see https://arxiv.org/abs/1810.09380).

Graph products are a class of groups that 'interpolate' between direct and free products, and generalise the notion of right-angled Artin groups. Given a property that free products (and maybe direct products) are known to satisfy, a natural question arises: do graph products satisfy this property? For instance, it is known that graph products act on tree-like spaces (quasi-trees) in a nice way (acylindrically), just like free products. In the talk we will discuss a construction of such an action and, if time permits, its relation to solving systems of equations over graph products.

Cubulating a group means finding a proper cocompact action on a CAT(0) cube complex. I will describe how cubulating a group tells us some nice properties of the group, and explain a general strategy for finding cubulations.

We will study the l1-homology of the 2-class in one relator groups. We will see that there are many qualitative and quantitive similarities between the l1-norm of the top dimensional class and the stable commutator length of the defining relation. As an application we construct manifolds with small simplicial volume.

This work in progress is joint with Clara Loeh.

One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This generalises work of Whyte and Mosher--Sageev--Whyte.

In the 80s, Hatcher and Thurston introduced the pants graph as a tool to prove that the mapping class group of a closed, orientable surface is finitely presented. The pants graph remains relevant for the study of the mapping class group, sitting between the marking graph and the curve graph. More precisely, there is a sequence of natural coarse lipschitz maps taking the marking graph via the pants graph to the curve graph.

A second motivation for studying the pants graph comes from Teichmüller theory. Brock showed that the pants graph can be interpreted as a combinatorial model for Teichmüller space with the Weil-Petersson metric.

In this talk I will introduce the pants graph, discuss some of its properties and state a few open questions.