Past Junior Topology and Group Theory Seminar

2 December 2015
16:00
Nicolaus Heuer
Abstract

Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.

We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as

  • the image $H$  hyperbolic groups,
  • the image $H$ discrete rank one isometries,
  • the preimage $G$ cyclic / free group, etc.

Furthermore, we point out a relation between QHM and extensions by short
exact sequences.

  • Junior Topology and Group Theory Seminar
25 November 2015
16:00
Federico Vigolo
Abstract

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.

  • Junior Topology and Group Theory Seminar
18 November 2015
16:00
Kieran Calvert
Abstract

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.

  • Junior Topology and Group Theory Seminar
11 November 2015
16:00
Robert Kropholler
Abstract

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. 

  • Junior Topology and Group Theory Seminar
4 November 2015
16:00
Giles Gardam
Abstract

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.

 

  • Junior Topology and Group Theory Seminar
28 October 2015
16:00
Ainhoa Iniguez
Abstract

 

Let $G$ be a finite group and let $w$ be a word in $k$ variables. We write $P_w(g)$ the probability that a random tuple $(g_1,\ldots,g_k)\in G^{(k)}$ satisfies $w(g_1,\ldots,g_k)=g$. For non-solvable groups, it is shown by Abért that $P_w(1)$ can take arbitrarily small values as $n\rightarrow\infty$. Nikolov and Segal prove that for any finite group, $G$ is solvable if and only if $P_w(1)$ is positively bounded from below as $w$ ranges over all words. And $G$ is nilpotent if and only if $P_w(g)$ is positively bounded from below as $w$ ranges over all words that represent $g$Alon Amit conjectured  that in the specific case of finite nilpotent groups and for any word, $P_w(1)\ge 1/|G|$.
 
We can also consider $N_w(g)=|G|^k\cdot P_w(g)$, the number of solutions of $w=g$ in $G^{(k)}$. Note that $N_w$ is a class function. We prove that if $G$ is a finite $p$-group of nilpotency class 2, then $N_w$ is a generalized character. What is more, if $p$ is odd, then $N_w$ is a character and for $2$-groups we can characterize when $N_{x^{2r}}$ is a character. What is more, we prove the conjecture of A. Amit for finite groups of nilpotency class 2. This result was indepently proved by M. Levy. Additionally, we prove that for any word $w$ and any finite $p$-group of class two and exponent $p$, $P_w(g)\ge 1/|G|$ for $g\in G_w$. As far as we know, A. Amit's conjecture is still open for higher nilpotency class groups. For $p$-groups of higher nilpotency class, we find examples of words $w$ for which $N_w$ is no longer a generalized character. What is more, we find examples of non-rational words; i.e there exist finite $p$-groups $G$ and words $w$ for which $g\in G_w$ but $g^{i}\not\in G_w$ for some $(i,p)=1$.

  • Junior Topology and Group Theory Seminar
21 October 2015
16:00
Alexander Margolis
Abstract

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.

  • Junior Topology and Group Theory Seminar
17 June 2015
16:00
Alejandra Garrido Angulo
Abstract

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.

  • Junior Topology and Group Theory Seminar
10 June 2015
16:00
Nina Otter
Abstract

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.

  • Junior Topology and Group Theory Seminar
3 June 2015
16:00
Robert Kropholler
Abstract

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 $n$.

  • Junior Topology and Group Theory Seminar

Pages