Past Algebra Seminar

4 February 2014
17:00
Tim Riley
Abstract
For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the defining relations. A "hydra phenomenon" gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. I will explain why, nevertheless, there are efficient (polynomial time) solutions to the Word Problems of these groups. The main innovation is a means of computing efficiently with compressed forms of enormous integers. This is joint work with Will Dison and Eduard Einstein.
10 December 2013
17:00
Richard Weidmann
Abstract
We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).
19 November 2013
17:00
Francesco Matucci
Abstract
Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties. This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.
12 November 2013
17:00
Ben Martin
Abstract
Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the number of isomorphism classes of irreducible $n$-dimensional complex characters of $\Gamma$. Representation growth is the study of the behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of representation growth. We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is bounded by a polynomial in $n$. I will discuss a question posed by Brent Everitt: can a group with polynomial representation growth have the alternating group $A_n$ as a quotient for infinitely many $n$?
5 November 2013
17:00
Andrei Jaikin-Zapirain
Abstract
I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez: For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.
22 October 2013
17:00
Gunnar Traustason
Abstract
Let F be a field. A symplectic alternating algebra over F consists of a symplectic vector space V over F with a non-degenerate alternating form that is also equipped with a binary alternating product · such that the law (u·v, w)=(v·w, u) holds. These algebraic structures have arisen from the study of 2-Engel groups but seem also to be of interest in their own right with many beautiful properties. We will give an overview with a focus on some recent work on the structure of nilpotent symplectic alternating algebras.

Pages