Forthcoming events in this series
I will talk about the boundaries of CAT(0) groups giving definitions, some examples and will state some theorems. I may even prove something if there is time.
I will discuss a couple of techniques often useful to prove quasi-isometric rigidity results for isometry groups. I will then sketch how these were used by B. Kleiner and B. Leeb to obtain quasi-isometric rigidity for the class of fundamental groups of closed locally symmetric spaces of noncompact type.
We will discuss various familiar properties of groups studied in geometric group theory, whether or not they are invariant under quasi-isometry, and why.
One can ask whether the fundamental groups of 3-manifolds are distinguished by their sets of finite quotients. I will discuss the recent solution of this question for Seifert fibre spaces.
I will illustrate how to build families of expanders out of 'very mixing' actions on measure spaces. I will then define the warped cones and show how these metric spaces are strictly related with those expanders.
Quasihomomorphisms (QHMs) are maps $f$ between groups such that the
homomorphic condition is boundedly satisfied. The case of QHMs with
abelian target is well studied and is useful for computing the second
bounded cohomology of groups. The case of target non-abelian has,
however, not been studied a lot.
We will see a technique for classifying QHMs $f: G \rightarrow H$ by Fujiwara and
Kapovich. We will give examples (sometimes with proofs!) for QHM in
various cases such as
Furthermore, we point out a relation between QHM and extensions by short
exact sequences.
A family of expanders is a sequence of finite graphs which are both sparse and highly connected. Firstly defined in the 80s, they had huge applications in applied maths and computer science. Moreover, it soon turned out that they also had deep implications in pure maths. In this talk I will introduce the expander graphs and I will illustrate a way to construct them by approximating actions of groups on probability spaces.
In 1964 Golod and Shafarevich discovered a powerful tool that gives a criteria for when a certain presentation defines an infinite dimensional algebra. In my talk I will assume the main machinery of the Golod-Shafarevich inequality for graded algebras and use it to provide counter examples to certain analogues of the Burnside problem in infinite dimensional algebras and infinite groups. Then, time dependent, I will define the Tarski number for groups relating to the Banach-Tarski paradox and show that we can using the G-S inequality show that the set of Tarski numbers is unbounded. Despite the fact we can only find groups of Tarski number 4, 5 and 6.
I will discuss a notoriously hard problem in group theory known as the flat closing conjecture. This states that a group with a finite classifying space is either hyperbolic or contains a Baumslag-Solitar Subgroup. I will give some strategies to try and create a counterexample to this conjecture.
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.
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.
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.
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.
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 $n$.
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.
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.
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.
Localization and completion of spaces are fundamental tools in homotopy theory. "Zabrodsky mixing" uses localization to "mix homotopy types". It was used to provide a counterexample to the conjecture that any finite H-space which is $A_3$ is also $A_\infty$. The material in this talk will be very classical (and rather basic). I will describe Sullivan's localization functor and demonstrate Zabrodsky's mixing by constructing a non-classical H-space.
A group is called residually finite if every non-trivial element can be homomorphically mapped to a finite group such that the image is again non-trivial. Residually finite groups are interesting because quite a lot of information about them can be reconstructed from their finite quotients. Baumslag showed that if G is a finitely generated residually finite group then Aut(G) is also residually finite. Using a similar method Grossman showed that if G is a finitely generated conjugacy separable group with "nice" automorphisms then Out(G) is residually finite. The graph product is a group theoretic construction naturally generalising free and direct products in the category of groups. We show that if G is a finite graph product of finitely generated residually finite groups then Out(G) is residually finite (modulo some technical conditions)
A Kähler group is a group which is isomorphic to the fundamental group of a compact Kähler manifold. In 2008 Dimca and Suciu proved that the groups which are both Kähler and isomorphic to the fundamental group of a closed 3-manifold are precisely the finite subgroups of $O(4)$ which act freely on $S^3$. In this talk we will explain Kotschick's proof of this result. On the 3-manifold side the main tools that will be used are the first Betti number and Poincare Duality and on the Kähler group side we will make use of the Albanese map and some basic results about Kähler groups. All relevant notions will be explained in the talk.
In Bass-Serre theory, one derives structural properties of groups from their actions on simplicial trees. In this talk, we introduce the theory of groups acting on $\mathbb{R}$-trees. In particular, we explain how the Rips machine is used to classify finitely generated groups which act freely on $\mathbb{R}$-trees.
I will look at some decidability questions for subgroups of Aut($F_n$) for general $n$. I will then discuss semisimple actions of Aut($F_n$) on complete CAT(0) spaces proving that the Nielsen moves will act elliptically. I will also look at proving Aut($F_3$) is large and if time permits discuss the fact that Aut($F_n$) is not Kähler
This talk will give an almost complete proof of the h-cobordism theorem, paying special attention to the sources of the dimensional restrictions in the theorem. If time allows, the alterations needed to prove its cousin, the s-cobordism theorem, will also be sketched.