A family of expanders is a sequence of finite graphs which are both sparse and highly connected. Firstly defined in the 80s, they had huge applications in applied maths and computer science. Moreover, it soon turned out that they also had deep implications in pure maths. In this talk I will introduce the expander graphs and I will illustrate a way to construct them by approximating actions of groups on probability spaces.

# Past Junior Topology and Group Theory Seminar

In 1964 Golod and Shafarevich discovered a powerful tool that gives a criteria for when a certain presentation defines an infinite dimensional algebra. In my talk I will assume the main machinery of the Golod-Shafarevich inequality for graded algebras and use it to provide counter examples to certain analogues of the Burnside problem in infinite dimensional algebras and infinite groups. Then, time dependent, I will define the Tarski number for groups relating to the Banach-Tarski paradox and show that we can using the G-S inequality show that the set of Tarski numbers is unbounded. Despite the fact we can only find groups of Tarski number 4, 5 and 6.

I will discuss a notoriously hard problem in group theory known as the flat closing conjecture. This states that a group with a finite classifying space is either hyperbolic or contains a Baumslag-Solitar Subgroup. I will give some strategies to try and create a counterexample to this conjecture.

This talk will be an easy introduction to some CAT(0) geometry. Among other things, we'll see why centralizers in groups acting geometrically on CAT(0) spaces split (at least virtually). Time permitting, we'll see why having a geometric action on a CAT(0) space is not a quasi-isometry invariant.

For a finitely generated group $G$ with subgroup $H$ we define $e(G,H)$, the relative ends of the pair $(G,H)$, to be the number of ends of the Cayley graph of G quotiented out by the left action of H. We will examine some basic properties of relative ends and will outline the theorem of Sageev showing that $e(G,H)>1$ if and only if $G$ acts essentially on a simply connected CAT(0) cube complex. If time permits, we will outline Niblo's proof of Stallings' theorem using Sageev's construction.

In this talk, we will introduce the notions of systolic and residual girth growth for finitely generated groups. We will explore the relationship between these types of growth and the usual word growth for finitely generated groups.

Associahedra are polytopes introduced by Stasheff to encode topological semigroups in which associativity holds up to coherent homotopy. These polytopes naturally form a topological operad that gives a resolution of the associative operad. Muro and Tonks recently introduced an operad which encodes $A_\infty$ algebras with homotopy coherent unit.

The material in this talk will be fairly basic. I will cover operads and their algebras, give the construction of the $A_\infty$ operad using the Boardman-Vogt resolution, and of the unital associahedra introduced by Muro and Tonks.

Depending on time and interest of the audience I will define unital $A_\infty$ differential graded algebras and explain how they are precisely the algebras over the cellular chains of the operad constructed by Muro and Tonks.

I will discuss various types of filling functions on topological spaces, stating some results in the area. I will then go onto prove that a finitely presented subgroup of a hyperbolic group of cohomological dimension 2 is hyperbolic. On the way I will prove a stronger result about filling functions of subgroups of hyperbolic groups of cohomological dimension $n$.