Forthcoming events in this series
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.
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).
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.
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.
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.
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.
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.
... for Torelli groups of surfaces.
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 $\mathrm{Out}(F_n)$.
First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.
We begin by showing the underlying ideas Bourgain used to prove that the Cayley graph of the free group of finite rank can be embedded into a Hilbert space with logarithmic distortion. Equipped with these ideas we then tackle the same problem for other metric spaces. Time permitting these will be: amalgamated products and HNN extensions over finite groups, uniformly discrete hyperbolic spaces with bounded geometry and Cayley graphs of cyclic extensions of small cancellation groups.
We'll discuss 2 ways to decompose a 3-manifold, namely the Heegaard
splitting and the celebrated geometric decomposition. We'll then see
that being hyperbolic, and more in general having (relatively)
hyperbolic fundamental group, is a very common feature for a 3-manifold.
A factorability structure on a group G is a specification of normal forms
of group elements as words over a fixed generating set. There is a chain
complex computing the (co)homology of G. In contrast to the well-known bar
resolution, there are much less generators in each dimension of the chain
complex. Although it is often difficult to understand the differential,
there are examples where the differential is particularly simple, allowing
computations by hand. This leads to the cohomology ring of hv-groups,
which I define at the end of the talk in terms of so called "horizontal"
and "vertical" generators.
We'll survey some of the ways that hyperbolic groups have been studied
using analysis on their boundaries at infinity.
I will give a brief introduction to the Steenrod squares and move on to show some applications of them in Topology and Geometry.
The construction of the asymptotic cone of a metric space which allows one to capture the "large scale geometry" of that space has been introduced by Gromov and refined by van den Dries and Wilkie in the 1980's. Since then asymptotic cones have mainly been used as important invariants for finitely generated groups, regarded as metric spaces using the word metric.
However since the construction of the cone requires non-principal ultrafilters, in many cases the cone itself is very hard to compute and seemingly basic questions about this construction have been open quite some time and only relatively recently been answered.
In this talk I want to review the definition of the cone as well as considering iterated cones of metric spaces. I will show that every proper metric space can arise as asymptotic cone of some other proper space and I will answer a question of Drutu and Sapir regarding slow ultrafilters.
After a quick-and-dirty introduction to nonstandard analysis, we will
define the asymptotic cones of a metric space and we will play around
with nonstandard tools to show some results about them.
For example, we will hopefully prove that any separable asymptotic cone
is proper and we will classify real trees appearing as asymptotic cones
of groups.
A brief survey of the above.
Geoghegan's stack construction is a tool for analysing groups
that act on simply connected CW complexes, by providing a topological
description in terms of cell stabilisers and the quotient complex,
similar to what Bass-Serre theory does for group actions on trees. We
will introduce this construction and see how it can be used to give
results on finiteness properties of groups.
This talk introduces the topic of random walks on a finitely generated group and asks what properties of such a group can be detected through knowledge of such walks.
Starting from a definition of the cohomology of a group, we will define the bounded cohomology of a group. We will then show how quasi-homomorphisms lead to cocycles in the second bounded cohomology group, and use this to look at the second bounded cohomology of some of our favourite groups. If time permits we will end with some applications.
I will introduce Symmetric spaces via a result of Kleiner & Leeb, comparing the axioms in their definition of a Euclidean building with properties of symmetric spaces of noncompact type.
Take a group G and split it as the fundamental group of a graph of groups, then take the vertex groups and split them as fundamental groups of graphs of groups etc. If at some point you end up with a collection of unsplittable groups, then you have a hierarchy. Haken showed that for any 3-manifold M with an incompressible surface S, one can cut M along S and and then find other incompressible surfaces in M\S and cut again, and repeating this process one eventually obtains a collection of balls. Analogously, Delzant and Potyagailo showed that for any finitely presented group without 2-torsion and a certain sensible class E of subgroups of G, G admits a hierarchy where the edge groups of the splittings lie in E. I really like their proof and I will present it.