Forthcoming events in this series


Wed, 31 Oct 2012

16:00 - 17:00
SR2

Bass-Serre Theory for Fusion Systems

Jason Semeraro
Abstract

Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.

Wed, 24 Oct 2012

16:00 - 17:00
L2

The metric geometry of mapping class groups

David Hume
Abstract

We give a brief overview of hyperbolic metric spaces and the relatively hyperbolic counterparts, with particular emphasis on the quasi-isometry class of trees. We then show that an understanding of the relative version of such spaces - quasi tree-graded spaces -  has strong consequences for mapping class groups. In particular, they are shown to embed into a finite product of (possibly infinite valence) simplicial trees. This uses and extends the work of Bestvina, Bromberg and Fujiwara.

Wed, 17 Oct 2012

16:00 - 17:00
SR2

Words and growth of groups acting on rooted trees

Elisabeth Fink
(University of Oxford)
Abstract

I will explain a construction of a group acting on a rooted tree, related to the Grigorchuk group. Those groups have exponential growth, at least under certain circumstances. I will also show how it can be seen that any two elements fulfil a non-trivial relation, implying the absence of non-cyclic free subgroups.

Wed, 10 Oct 2012

16:00 - 17:00
SR2

A Voyage into Outer Space (what I did on my holidays)

Henry Bradford
(University of Oxford)
Abstract

The study of free groups and their automorphisms has a long pedigree, going back to the work of Nielsen and Dehn in the early 20th century, but in many ways the subject only truly reached maturity with the introduction of Outer Space by Culler and Vogtmann in the “Big Bang” of 1986. In this (non-expert) talk, I will walk us through the construction of Outer Space and some related complexes, and survey some group-theoretic applications.

Wed, 26 Sep 2012

16:00 - 17:00
SR2

The concept of p-deficiency and its applications

Anitha Thillaisundaram
Abstract

We use Schlage-Puchta's concept of p-deficiency and Lackenby's property of p-largeness to show that a group having a finite presentation with p-deficiency greater than 1 is large. What about when p-deficiency is exactly one? We also generalise a result of Grigorchuk on Coxeter groups to odd primes.

Wed, 25 Apr 2012

16:00 - 17:00
SR2

Stabilisers of conjugacy classes in free groups under the action of automorphisms

Moritz Rodenhausen
Abstract

A construction by McCool gives rise to a finite presentation for the stabiliser of a finite set of conjugacy classes in a free group under the action of Aut(F_n) or Out(F_n). An important concept of my talk are rigid elements, which will allow to simplify these huge presentations. Finally I will sketch applications to centralisers in Aut(F_n).

Wed, 07 Mar 2012

16:00 - 17:00
SR2

TBA

John MacKay
Wed, 15 Feb 2012

16:00 - 17:00
SR2

Groups with prescribed finiteness properties

Stefan Witzel
Abstract

The first group known to be finitely presented but having infinitely generated 3rd homology was constructed by Stallings. Bieri extended this to a series of groups G_n such that G_n is of type F_{n-1} but not of type F_n. Finally, Bestvina and Brady turned it into a machine that realizes prescribed finiteness properties. We will discuss some of these examples.

Wed, 25 Jan 2012

16:00 - 17:00
SR2

Horocyclic Products, the Geometry of Some Metabelian Groups and Short Conjugators

Andrew Sale
Abstract

The lamplighter groups, solvable Baumslag-Solitar groups and lattices in SOL all share a nice kind of geometry. We'll see how the Cayley graph of a lamplighter group is a Diestel-Leader graph, that is a horocyclic product of two trees. The geometry of the solvable Baumslag-Solitar groups has been studied by Farb and Mosher and they showed that these groups are quasi-isometric to spaces which are essentially the horocyclic product of a tree and the hyperbolic plane. Finally, lattices in the Lie groups SOL can be seen to act on the horocyclic product of two hyperbolic planes. We use these spaces to measure the length of short conjugators in each type of group.

Wed, 18 Jan 2012

15:45 - 16:45
SR2

TBC

Owen Cotton-Barratt
Wed, 30 Nov 2011

16:00 - 17:00
SR2

Classical and Cubical Small Cancellation Theory

David Hume
Abstract

Building on the previous talk, we continue the exploration of techniques required to understand Wise's results. We present an overview of classical small cancellation theory running in parallel with the newer one for cubical complexes.

Wed, 23 Nov 2011

16:00 - 17:00
SR2

Cube Complexes and Quasi-Convex Hierarchies

David Hume
Abstract

We present recent results of Dani Wise which tie together many of the themes of this term's jGGT meetings: hyperbolic and relatively hyperbolic groups, (in particular limit groups), graphs of spaces, 3-manifolds and right-angled Artin groups.
Following this, we make an attempt at explaining some of the methods, beginning with special non-positively curved cube complexes.

Wed, 22 Jun 2011

16:00 - 17:00
SR1

Parallelogram Law for Isometries of CAT(0)-spaces

Moritz Rodenhausen
(Rheinische Friedrich-Wilhelms-Universität Bonn)
Abstract

In euclidean space there is a well-known parallelogram law relating the

length of vectors a, b, a+b and a-b. In the talk I give a similar formula

for translation lengths of isometries of CAT(0)-spaces. Given an action of

the automorphism group of a free product on a CAT(0)-space, I show that

certain elements can only act by zero translation length. In comparison to

other well-known actions this leads to restrictions about homomorphisms of

these groups to other groups, e.g. mapping class groups.

Wed, 15 Jun 2011

16:00 - 17:00
SR1

Cutting and pasting...

Martin Palmer
(University of Oxford)
Abstract

... for Torelli groups of surfaces.

Wed, 08 Jun 2011

16:00 - 17:00
SR1

Fusion, graphs and $\mathrm{Out}(F_n)$.

Dawid Kielak
(University of Oxford)
Abstract

We will attempt to introduce fusion systems in a way comprehensible to a Geometric Group Theorist. We will show how Bass--Serre thoery allows us to realise fusion systems inside infinite groups. If time allows we will discuss a link between the above and $\mathrm{Out}(F_n)$.