International Young Seminar on
Bounded Cohomology and Simplicial Volume 

Abstract:

L-infinity cohomology is a quasi-isometry invariant of finitely generated groups. It was introduced by Gersten as a tool to find lower bounds for the Dehn function of some finitely presented groups. I will discuss a generalisation of a theorem of Gersten on surjectivity of the restriction map in L-infinity cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and L-infinity cohomology calculations for some well-known classes of groups such as RAAGs, Bestvina-Brady groups and Out(F_n). Along the way, we obtain hyperbolicity criteria for groups of type FP_2(Q) and for those satisfying a rational homological linear isoperimetric inequality.

I will first define L-infinity cohomology and discuss some of its properties. Time permitting, I will then sketch some of the main ideas behind the proofs. This is joint work with Vladimir Vankov.

Abstract:

Racks and quandles are algebraic structures that solve the Yang-Baxter equation and appear naturally in Knot Theory. They are investigated from many points of view. In the talk I will introduce a natural metric for racks and relate this metric to a bi-invariant metric on the group of inner automorphism of the rack. I will show examples, where the metric has finite diameter as well as unbounded examples. I will also introduce bounded cohomology of a rack or a quandle and relate the unboundedness of the above metrics to the second bounded cohomology. If time permits, I will also discuss the amenability in this context.

Abstract:

Let Γ be a finitely generated group and consider a complete metric space X. An intriguing and at the same time very vague question is to understand how many isometric Γ-actions there are on X and if there exists a way to classify them. When X is an irreducible Hermitian symmetric space not of tube type, Burger, Iozzi and Wienhard showed that the existence of a Kahler form allows to classify Zariski dense Γ-representations in the isometry group G:=Isom(X). More precisely, since the integral of the Kahler form defines naturally a continuous bounded 2-class for G, Burger, Iozzi and Wienhard proved that its pullback along a Zariski dense representation is a complete invariant in the conjugacy class of the action.

We will show that something completely analogous holds in the context of representations of semidirect groupoids associated to a measure preserving action of Γ on an ergodic standard Borel probability space. As a consequence there are no such representations for both higher rank lattices and lattices in products. This is a joint work with Filippo Sarti.

Abstract:

We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. What we prove here is similar to the sharp lower bound 1/2 of stable commutator length in free products of torsion free groups (for hyperbolic elements), but the setting is different.

This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain injectivity of the map G->(G*Z)/<<w>> when w is a proper power for arbitrary G. Both results generalize to certain HNN extensions.

Abstract:

I will talk about joint work with Grigori Avramidi and Boris Okun where we construct closed, aspherical 7-manifolds with word hyperbolic, special fundamental group that do not virtually fiber over the circle. 

Abstract:

Stable commutator length (scl) is a measure of homological complexity in groups. Calegari's celebrated Rationality Theorem provides an algorithm to compute scl in free groups; this has been generalised to larger classes of groups, but computations of scl remain elusive in closed surface groups. In this talk, I will present some isometric embedding results for scl in fundamental groups of surfaces with boundary; I will then introduce the relative Gromov seminorm and explain how it allows one to obtain natural generalisations of results for scl in free groups to the possibly closed case.

Abstract:

To compute invariants of 3-manifolds in practice, we generally give them as triangulations. Two questions immediately arise: how large is the triangulation, and are there small representatives of the topology of the 3-manifold in it? The triangulation complexity of a 3-manifold is the minimal number of tetrahedra in a triangulation of it. This may sound reminiscent of integral simplicial volume, but has recently been shown to not be Lipshitz equivalent to it. I will give the approximate triangulation complexity of Seifert fibered spaces, which are a large class of 3-manifolds arising from geometric decomposition, by finding bounded complexity representatives of some interesting curves and surfaces in any triangulation of one.

Abstract:

In this talk we are going see why a large class of topological full groups acting on the Cantor set has vanishing bounded cohomology with respect to nice coefficients. In particular, this method shows bounded acyclicity of Thompson's group V as well as the full homeomorphism group of the Cantor set. We are going to use that acyclic resolutions calculate bounded cohomology and exploit the fact that these groups have good transitivity properties. Moreover, based on this approach we will discuss why Thompson's group T fails to be boundedly acyclic, whereas F is boundedly acyclic again.