• Topology
• Workshop: New directions in L2-invariants

• Schedule
• Participants

A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. We prove a vanishing result on the rank gradient and growth of torsion in the first homology for right angled groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. Using rigidity theory it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple Lie group. A joint work with Tsachik Gelander and Nikolay Nikolov.

Nicolas Bergeron: L²-invariants of locally symmetric spaces

The first lecture will focus on growth of Betti numbers of locally symmetric spaces. I will in particular recall the notion of L²-Betti numbers and explains how these "localize" in the locally homogeneous setting.

In the second lecture I will discuss attempts to study homology torsion growth with somewhat similar methods. In particular I will discuss analytic torsion and its L²-avatar.

Finally in the last lecture we will explain how to compute explicitly the L²-invariants discussed in the first two lectures.

The theory of unitary representations describes locally compact groups from the point of view of harmonic analysis and operator theory. To provide a good understanding there are various concepts that are considered. In this talk I present recent results and open questions regarding the Hecke algebras of (totally disconnected) locally compact groups in relation to the Howe-Moore property and type I groups.

Video recording

Stefan Friedl: The L²-Alexander function of knots and 3-manifolds

We will introduce the L²-Alexander function of knots and 3-manifolds and we will relate its connections to the geometric and topological properties of the underlying manifold.

Video - Lecture 1

Video - Lecture 2

Thanks to the Lück approximation theorem, the L²-Betti numbers of a normal covering of CW-complexes can be defined as limits of dimensions of the ordinary rational homology of intermediate covers. This leads to an interesting generalization of L²-Betti numbers: instead of the rational homology, one can try to use homology with coefficients in an arbitrary field k. This results in so called k-homology gradients. They were first suggested by Farber and later studied extensively by Lackenby and other authors. I will talk about a recent joint work with Thomas Schick where we study variants of the Atiyah problem for k-homology gradients. While overall results are similar to the results about L²-Betti numbers, there are some surprising differences: it is impossible to obtain an irrational k-homology gradient when the fundamental group of the base space is the lamplighter group, and the field k is of positive characteristic. We also disprove a conjecture of A. Thom, by showing that in general the k-homology gradients do not stabilize as the characteristic of k tends to infinity. Finally, we point out a family of concrete group ring elements in positive characteristic for which we currently do not know whether the Lück approximation holds.

Video recording

The universal L²-torsion, introduced by Friedl and Lück, allows for an extension of the Thurston norm from the setting of 3-manifolds to that of free-by-cyclic groups. We will discuss this extension, and show that this norm and the Alexander norm for F₂-by-ℤ satisfy an inequality analogous to the one satisfied by the Thurston and Alexander norms on 3-manifolds. We will also discuss the relationship between the universal L²-torsion and the Bieri-Neumann-Strebel invariants.

This is joint work with Florian Funke.

Video recording

Yi Liu: On the L²-Alexander torsion of 3-manifolds

For an irreducible orientable compact 3-manifold with empty or incompressible toral boundary, the full L²-Alexander torsion associated to any real first cohomology class of that manifold is represented by a function of a positive real variable. In this talk, I will discuss some ideas to show that the function is continuous, everywhere positive, and asymptotically monomial in both ends.

Video recording

Wolfgang Lück: Universal L²-torsion, L²-Euler characteristics, Thurston norms and polytopes

We assign to a finite CW-complex and a cocycle in its first cohomology a twisted version of the L²-Euler characteristic and study its main properties. In the case of an irreducible orientable 3-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. We discuss its relation to the degree of higher order Alexander polynomials in the sense of Cochran and Harvey.

All these invariants come from another invariant called universal L²-torsion, which can be related to the Grothendieck groups of integral polytopes in ℝⁿ under the Minkowski sum.

This is a joint project with Stefan Friedl.

Video recording

Suppose an amenable group G is acting freely on a simply connected simplicial complex X^~ with compact quotient X. Fix n ≥ 1, assume H_n(X^~, ℤ)=0 and let (H_i) be a Farber chain in G. We prove that the torsion of the integral homology in dimension n of X^~/H_i grows subexponentially in [G:H_i]. This fails if X is not compact. We provide the first examples of amenable groups for which torsion in homology grows faster than any given function. These examples include some solvable groups of derived length 3 which is the minimal possible.

Joint work with Aditi Kar and Peter Kropholler.

Video recording

I will give a survey on cohomological finiteness conditions for classifying spaces for families of subgroups, such as the dimension or the type and will discuss some old and new questions.

Thomas Schick: On the center-valued Atiyah conjecture for L²-Betti numbers

The so-called Atiyah conjecture states that the von Neumann dimensions of the L²-homology modules of free G-CW-complexes belong to a certain set of rational numbers, depending on the finite subgroups of G. In this article we extend this conjecture to a statement for the center-valued dimensions. We show that the conjecture is equivalent to a precise description of the structure as a semisimple Artinian ring of the division closure D(ℚG) of ℚ[G] in the ring of affiliated operators. We prove the conjecture for all groups in Linnell's class C, containing in particular free-by-elementary amenable groups.

The center-valued Atiyah conjecture states that the center-valued L²-Betti numbers of finite free G-CW-complexes are contained in a certain discrete subset of the center of ℂ[G], the one generated as an additive group by the center-valued traces of all projections in ℂ[H], where H runs through the finite subgroups of G.

Finally, we use the approximation theorem of Knebusch for the center-valued L²-Betti numbers to extend the result to many groups which are residually in C, in particular for finite extensions of products of free groups and of pure braid groups.

This is joint work with Anselm Knebusch and Peter Linnel.

Video recording