Past Algebra Seminar

27 May 2014
15:00
Dennis Dreesen
Abstract
The common convention when dealing with hyperbolic groups is that such groups are finitely generated and equipped with the word length metric relative to a fi nite symmetric generating subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the fi nitely generated setting. We study the class of locally compact hyperbolic groups and elaborate on the similarities and diff erences between the discrete and non-discrete setting.
13 May 2014
17:00
Eric Swenson
Abstract
Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and $f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$. We prove that the join of two Cantor sets and its suspension are Tits rigid.
6 May 2014
17:00
Alain Valette
Abstract
A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.
29 April 2014
17:00
Abstract
A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$ is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a so called state-closed (or, self-similar in dynamical terms) representation of $G$ on a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$; i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant. Examples of state-closed groups are the Grigorchuk 2-group and the Gupta- Sidki $p$-groups in their natural representations on rooted trees. The affine group $Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed representations. Yet another example is the free nilpotent group $G = F (c; d)$ of class c, freely generated by $x_i (1\leq  i \leq  d)$: let $H = \langle x_i^n | \ (1 \leq  i \leq  d) \rangle$ where $n$ is a fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i \rightarrow x_i$ $(1 \leq  i \leq  d)$. We will discuss state-closed representations of general abelian groups and of …nitely generated torsion-free nilpotent groups.
4 March 2014
17:00
Dr David Craven
Abstract
The maximal subgroups of the exceptional groups of Lie type have been studied for many years, and have many applications, for example in permutation group theory and in generation of finite groups. In this talk I will survey what is currently known about the maximal subgroups of exceptional groups, and our recent work on this topic. We explore the connection with extending morphisms from finite groups to algebraic groups.
25 February 2014
17:00
Abstract
We define the notion of orbit decidability in a general context, and descend to the case of groups to recognise it into several classical algorithmic problems. Then we shall go into the realm of free groups and shall analise this notion there, where it is related to the Whitehead problem (with many variations). After this, we shall enter the negative side finding interesting subgroups which are orbit undecidable. Finally, we shall prove a theorem connecting orbit decidability with the conjugacy problem for extensions of groups, and will derive several (positive and negative) applications to the conjugacy problem for groups.
18 February 2014
17:00
Chris Parker
Abstract
In this talk, I will explain part of the programme of Gorenstein, Lyons and Solomon (GLS) to provide a new proof of the CFSG. I will focus on the difference between the initial notion of groups of characteristic $2$-type (groups like Lie type groups of characteristic $2$) and the GLS notion of groups of even type. I will then discuss work in progress with Capdeboscq to study groups of even type and small $2$-local odd rank. As a byproduct of the discussion, a picture of the structure of a finite simple group of even type will emerge.

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