Past 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
20 May 2015
16:00
Federico Vigolo
Abstract

In this talk I will try to show how certain asymptotic properties of a random walk on a graph are related to geometric properties of the graph itself. A special focus will be put on spectral properties and isoperimetric inequalities, proving Kesten's criterion for amenability.

  • Junior Topology and Group Theory Seminar
13 May 2015
16:00
Alexander Margolis
Abstract

We say a group is accessible if the process of iteratively decomposing G as an amalgamated free product or HNN extension over a finite group terminates in a finite number of steps. We will see Dunwoody's proof that FP2 groups are accessible, but that finitely generated groups need not be. If time permits, we will examine generalizations by Bestvina-Feighn, Sela and Louder.

  • Junior Topology and Group Theory Seminar
6 May 2015
16:00
Giles Gardam
Abstract

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.

  • Junior Topology and Group Theory Seminar

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