A stacking is a lift of an immersion of graphs $A\to B$ to an embedding of $A$ into the product of $B$ with the real line; their existence relates to orderability properties of groups. I will describe how Louder and Wilton used them to prove Wise's "$w$-cycles" conjecture: given a primitive word $w$ in a free group $F$, and a subgroup $H < F$, the number of conjugates of $H$ which intersect $<w>$ nontrivially is at most rank($H$). I will also discuss applications of the result to questions of coherence, and possible extensions of it.

# Past Junior Topology and Group Theory Seminar

I will give a self-contained introduction to the theory of cross ratios on boundaries of Gromov hyperbolic and CAT(-1) spaces, focussing on the connections to the following two questions. When are two spaces with the 'same' Gromov boundary isometric/quasi-isometric? Are closed Riemannian manifolds completely determined (up to isometry) by the lengths of their closed geodesics?

When groups may be built up as graphs of 'simpler' groups, it is often

of interest to study how good residual finiteness properties of simpler

groups can imply residual properties of the whole. The essential case of

this theory is the study of residual properties of finite groups. In

this talk I will discuss the question of when a graph of finite

$p$-groups is residually $p$-finite, for $p$ a prime. I describe the

previous theorems in this area for one-edge and finite graphs of groups,

and their method of proof. I will then state my recent generalisation of

these theorems to potentially infinite graphs of groups, together with

an alternative and more natural method of proof. Finally I will briefly

describe a usage of these results in the study of accessibility --

namely the existence of a finitely generated inaccessible group which is

residually $p$-finite.

I will give a description of a method introduced by N. Ivanov to study the abstract commensurator of a group by using a rigid action of that group on a graph. We will sketch Ivanov's theorem regarding the abstract commensurator of a mapping class group. Time permitting, I will describe how these methods are used in some of my recent work with Horbez on outer automorphism groups of free groups.

Knot theory investigates the many ways of embedding a circle into the three-dimensional sphere. The study of these embeddings is not only important for understanding three-dimensional manifolds, but is also intimately related to many new and surprising phenomena appearing in dimension four. I will discuss how four-dimensional interpretations of some invariants can help us understand surfaces that bound a given link (embedding of several disjoint circles).

In 1982, Gromov introduced bounded cohomology to give estimates on the minimal volume of manifolds. Since then, bounded cohomology has become an independent and active research field. In this talk I will give an introduction to bounded cohomology, state many open problems and relate it to other fields.

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.

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.