Titles and abstracts

Mini-Courses

Wolfgang Lück (Universität Bonn)
1.) Basic Introduction to $L^2$-invariants

We give an elementary algebraic oriented introduction to $L^2$-Betti numbers including their basic properties and first applications.  We will also explain that the vanishing of the $L^2$-Betti numbers is an obstruction to (virtually) fibering over the circle.
We will briefly also introduce a secondary $L^2$-invariant given by $L^2$-torsion.  Note  that $L^2$-Betti numbers and the $L^2$-torsion are $L^2$-versions of the classical invariants given by Betti numbers and Reidemeister torsion.
The definition of the $L^2$-versions is harder but their behaviour is much better than in the classical case.
The main property of these $L^2$-invariant is that they can be used to solve problems in algebra, geometry, group theory, topology, and operator theory, where one would not expect them to show up at all.
This talk will be held on a rather elementary level.

2.) Prominent Conjectures and Applications of $L^2$-invariants

We will discuss basic conjectures about L^2-Betti numbers such as the Atyiah-Conjecture, the Singer Conjecture, Approximating Conjecture for Fuglede-Kadison determinants, and the one on homological growth and describe their status. We will discuss further applications, in particular to 3-manifolds.

3.) $L^2$-invariants and fibering over $S^1$: secondary obstructions

This will be an advanced research talk about an ongoing project with Sam Hughes. Suppose that the aspherical closed manifold $M$ has the property that all its $L^2$-Betti numbers over any field vanish. This is a necessary but not sufficient condition for $M$ to fiber over $S^1$. 

We will give two types of new secondary obstructions for fibering over $S^1$ which are based on twisted or universal $L^2$-torsion and some input from algebraic $K$-theory. 

Bruno Martelli (Università di Pisa)
Pseudo-Anosov homeomorphisms

By a famous theorem of Thurston, a fibering 3-manifold is hyperbolic if and only if the monodromy may be represented by a pseudo-Anosov homeomorphism. This is a particularly beautiful kind of self-homeomorphism of a surface, with various extremal dynamical properties. We propose a generalisation of pseudo-Anosov homeomorphism that works in every even dimension, study its properties, and show that there is a fibering hyperbolic 5-manifold (almost certainly the one constructed with Italiano and Migliorini) whose monodromy is of this kind. We study this example in detail. 

Research Talks

Macarena Arenas (University of Cambridge / ICMAT)
Biautomaticity and cubical-small cancellation 

We show that under suitable hypotheses, (small-cancellation) quotients of cubulated groups are biautomatic. Based on joint work with Claudio Llosa-Isenrich.

Grigori Avramidi (MPIM Bonn)
Cell structures on hyperbolic manifolds

I'll talk about lower bounds on the number of cells in a cell structure on a hyperbolic manifold, expressed in terms of the injectivity radius of the manifold. Joint work with Thomas Delzant.

Sam Fisher (ICMAT Madrid)
Novikov homology and cohomology

We define the Novikov ring associated to an epimorphism f : G->Z and study the Novikov homology and cohomology of G, that is, the (co)homology of G with coefficients in the Novikov ring. We discuss applications that the vanishing of this (co)homology has for properties of ker(f), such as its finiteness properties and cohomological dimension. Partially based on joint works with Dawid Kielak and Giovanni Italiano and with Pablo Sánchez-Peralta.

Sam Hughes (Universität Bonn)
Insufficiency of fibring obstructions

In this talk I will present a number of examples showing that the vanishing of various homological obstructions to fibring a group or manifold over the circle are insufficient to conclude fibring.  If time permits I will present some new secondary obstructions.  Based on joint works with (some subsets of) Ian Leary, Marco Linton, and Wolfgang Lück.

Monika Kudlinska (University of Cambridge)
Thurston norm in groups

The Thurston norm in the setting of 3-manifolds measures the minimal topological complexity of an embedded surface dual to a given integral character. Given a finitely presented group G, it is sometimes possible to define an analogous function on the set of integral characters of G, which measures the minimal complexity of dual HNN splittings over finitely presented subgroups. We prove that if G is free-by-cyclic or the fundamental group of an admissible 3-manifold then such a function extends to a seminorm on the first cohomology of G with real coefficients. To do so, we prove that the complexity function coincides with the twisted L2-Euler characteristic. The main new technical tool is L2-subgroup rigidity for such groups. This is based on joint work with Andrei Jaikin-Zapirain and Pablo Sanchez-Peralta.

 
Claudio Llosa Isenrich (University of Luxembourg)
Simple groups separated by homological finiteness properties

A group is simple if it has no non-trivial quotients. Simple groups have long played a distinguished role in group theory. Recently, the geometry of simple groups and their finiteness properties have received a lot of attention, for instance through their role in the Boone--Higman Conjecture. The homotopical finiteness properties $F_n$ and the homological finiteness properties $FP_n(R)$, where $R$ is a unital abelian ring, generalize finite generation (which is equivalent to both $F_1$ and $FP_1(R) for any $R$) and finite presentability (which is equivalent to $F_2$). Skipper, Witzel and Zaremsky proved that for every $n\geq 0$ there is a simple group of type $F_n$ and not $F_{n+1}$. This raises the question of how varied the homological finiteness properties of infinitely presented simple groups can be. In this talk, I will explain a construction of examples of simple groups with the same homotopical and homological finiteness properties as Bestvina—Brady groups. In particular, our examples imply the existence of a simple group of type $FP_2(\mathbb{Z})$ which is not finitely presented, answering a question of Zaremsky. This is joint work with Eduard Schesler and Xiaolei Wu.

Armando Martino (University of Southampton)
The conjugacy problem in cyclic extensions of one ended hyperbolic groups.

Given a torsion-free, 1-ended hyperbolic group G and an automorphism f of G, one can form the mapping torus of G with monodromy f - call this M_f. We show that this has solvable conjugacy problem.
This builds on the work of Preaux, who proved that all compact (geometrisable) 3-manifolds have solvable conjugacy problem. Indeed, if G is a surface group, then M_f is a fibred 3-manifold group. Our point of view is, for the general case, to view M_f as analogous to a fibred 3-manifold and to construct what we call the `block decomposition' of M_f, which is a JSJ-type decomposition of it. This analogy will be explored in the talk.
Closely modelling the proof strategy of Preaux, we have a general criterion for when a graph of groups decomposition admits a solvable conjugacy problem and deduce that the block decomposition satisfies this criterion for M_f. In fact, when M_f is an 3-manifold group, the block decomposition is the geometric JSJ albeit allowing for non-orientability; we cut along both tori and Klein bottles.
If time allows, we will discuss potential extensions and generalisations of this result.  

Matteo Migliorini (Karlsruher Institut für Technologie)
F_2-fibring of high-dimensional hyperbolic groups

In this talk, we present a construction that produces hyperbolic groups in every cohomological dimension $d >= 3$ that fiber with finitely presented kernel.
These kernels arise from performing the combinatiorial game of Jankiewicz, Norin, and Wise on certain right-angled Coxeter groups. Our procedure is inspired by a similar construction of Lafont, Minemyer, Sorcar, Stover, and Wells, which only produces finitely generated kernels.
This is joint work with Italiano and Ng.

Boris Okun (University of Wisconsin–Milwaukee)
L^2 Betti numbers of RACG's based on surfaces

Let L be a flag triangulation of a closed genus g surface and W_L be the associated right-angled Coxeter group. The Davis complex in this case is a cubical 3-dimensional pseudo-manifold, with links of vertices isomorphic to L.
Mike Davis and I conjectured that L^2 Betti numbers of such W_L are concentrated in degree 2 and hence the second one equals to g. (For triangulations of a 2-sphere, this is just a special case of the Singer conjecture.)
I will describe the combinatorial side of an argument proving this for triangulations of tori.
Based on a joint work with Grigori Avramidi and Kevin Schreve. 

Kevin Schreve (Louisiana State University)
Edge subdivisions/contractions and right-angled Coxeter groups

Lutz and Nevo showed that two PL-homeomorphic flag complexes are related by a sequence of edge subdivisions and their inverses. A flag complex determines a right-angled Coxeter group (RACG), and in earlier work with Avramidi-Okun we showed that the L^2-Betti numbers of the RACG often decrease after edge subdivison. I will discuss a similar inequality for the  L^2-Betti numbers of the RACG after edge contraction, which is more general than inverting edge subdivision, and give some applications. Joint with Grigori Avramidi and Boris Okun.