HIM Trimester Program on

Rigidity

Workshop: Rigidity in cohomology, K-theory, geometry and ergodic theory

Date: November 23 - 27

Venue: HIM, Poppelsdorfer Allee 45

Exception: Only colloquium tea and final lecture by Wolfgang Lück will be held in kl. Hörsaal, Wegelerstraße 10 (See page bottom for the map).

Rigiditiy Workshop Participants

Schedule

Abstracts


Tim Austin: A counterexample to a conjecture of Atiyah
A 1972 conjecture of Atiyah asserts that for any finitely-generated group, the L^2 Betti numbers of any cocompact free proper \G-manifold should be rational numbers. Building on earlier work that converts this into a question about the dimension of the kernel of an element of the rational group ring of \G, I will describe a recent construction of a family of groups and group ring elements that provides counterexamples to this conjecture.

Arthur Bartels: Topological rigidity and non-positively curved groups
In this talk the connection between topological rigidity of aspherical manifolds and non-positively curved groups will be discussed.

Dietmar Bisch: Subfactors and rigidity
I will discuss some of the surprising algebraic structures which arise naturally in the theory of subfactors. In particular, I will explain the appearance of certain exotic fusion categories.

Martin Bridson: Actions, representations, and rigidity for Out(F) and mapping class groups
I shall discuss several results and open questions around rigidity for automorphism groups of free groups and mapping class groups. In particular I shall discuss actions of these groups on CAT(0) spaces as well their linear and non-linear actions on Euclidean spaces. And I shall explain a result of Piggott concerning virtual representations of Aut(F).

Marc Burger: On some recent advances in higher Teichmueller theory
This is a report on contributions of various people to the study of representations of the fundamental group of a compact surface into Lie groups, mainly of hermitian type. In particular we will discuss the class of positive representations proposed by Ben Simon and Hartnick, and which extends the notion of maximal representation.

Pierre-Emmanuel Caprace: A lattice in a product of more than two Kac-Moody groups is arithmetic
Margulis´ seminal works imply that irreducible lattices in semi-simple linear groups of rank at least two are (superrigid and) arithmetic. Burger-Mozes lattices in the automorphism group of a product of two trees, and Kac-Moody lattices in the automorphism group of a product of two buildings, provide examples of (superrigid) non-arithmetic lattices beyond the linear world. One expects however that the existence of non-arithmetic lattices should be a uniformly low rank phenomenon. The purpose of this talk is to explain that an irreducible lattice in a product of at least three Kac-Moody buildings of simply laced type is necessarily arithmetic. This is a joint work with Nicolas Monod.

Diarmuid Crowley: An introduction to the Manifold Atlas
I shall give a short introduction to the goals and structures of the Manifold Atlas, an on-line resource for information about manifolds being hosted at the HIM: http://www.manifoldatlas.him.uni-bonn.de The Atlas is a Wiki and part of the talk will cover first steps for using Wikis.

Jim Davis: Torus bundles over lens spaces
Let G be the semidirect product of Z/p acting on Z^n where the action is free away from 0. Then G is the fundamental group of a torus bundle over a lens space. We compute the homology of R^n/G, the topological K-theory of BG and the K-theory of the reduced C*-algebra of G. For the latter we use the Baum-Connes conjecture.
On the geometric side, we prove the Gromov-Lawson-Rosenberg Conjecture characterizing which Spin manifolds with fundamental group G admit metrics of positive scalar curvature. We compute the structure set of manifold homotopy equivalent to the torus bundle over lens spaces. This uses the Farrell-Jones Conjecture and is the first classification of manifolds whose fundamental group is not built from torsion-free groups and finite groups using amalgamated free products and HNN extensions.

This is joint work with Wolfgang Lueck.

Steve Ferry: Characterizing strange homology manifolds
In joint work with Bryant, Mio, and Weinberger, the author constructed examples of ANR homology manifolds of dimension >5 that satisfy the 2-dimensional disjoint disks property and that are nowhere euclidean, giving counterexamples to a conjecture of Cannon.
In this talk, I will show that all such homology manifolds can be obtained by means of our construction. This is joint work with Bryant and Mio.

Tsachik Gelander: On the dynamics of Aut(F_n) on character varieties
Little is known in general about the dynamics of the natural Aut(F_n) action on Hom(F_n,G) when G is a group carrying some additional structure (e.g. finite/compact/algebraic/...). I will describe some recent progress, and in particular a joint work of Y. Minsky and myself which shows that the action on the redundant part is always ergodic and minimal but not always week mixing, confirming a conjecture of A. Lubotzky. This is new already for G=SL(2,R) and some related interesting questions (about SL(2,R)) are still unknown.

Anders Karlsson: Building characters
I will discuss a few constructions of homomorphisms from finitely generated groups into abelian groups and applications to harmonic functions, entropy inequalities and drift of random walks. Based on joint works with Anna Erschler and with Francois Ledrappier.

Yoshikata Kida: Measure equivalence rigidity of amalgamated free products
Measure equivalence is an equivalence relation between discrete countable groups, defined in measure-theoretic terms. It is known that higher rank lattices and mapping class groups satisfy rigidity in the sense of measure equivalence. The latter groups G is indeed ME rigid, that is, any group ME to G is virtually isomorphic to G. This talk presents a construction of ME rigid groups given by amalgamated free products of two rigid groups.

Clara Löh: Manifolds and groups not presentable by products
The class of oriented closed connected manifolds of a given dimension is partially ordered by the domination relation, i.e., the relation given by the existence of maps of non-zero degree.
In this talk, we will concentrate on the question of which manifolds are not dominated by a non-trivial product of manifolds. For rationally essential manifolds, we give a sufficient condition for non-presentability by products in terms of the fundamental groups. E.g., groups containing a large amount of negative curvature are not presentable by products. We will discuss classes of groups that are not presentable by products, and study the relation between presentability by products and certain invariants from geometric and measurable group theory.

This is joint work with Dieter Kotschick.

Wolfang Lück: On hyperbolic groups with spheres as boundary
Let G be a torsion-free hyperbolic group and let n be an integer greater or equal to six. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere. This answers a question of Gromov.

Nicolas Monod: Product groups acting on manifolds
In joint work with Furman, we find obstructions for product groups to act on manifolds. To this end, we establish a cocycle super-rigidity theorem for products.

Narutaka Ozawa: Quasi-homomorphism rigidity with noncommutative targets
As a strengthening of Kazhdan´s property (T), property (TT) was introduced by Burger and Monod. In this talk, I will add more rigidity to (TT) and introduce property (TTT). This property is suited for the study of rigidity phenomena for quasi-homomorphisms with noncommutative targets and ε-representations.

Irine Peng: Quasi-isometric rigidities in solvable groups
Building on the works of Farb-Mosher and Eskin-Fisher-Whyte, I will describe some rigidity phenomena in the settings of HNN extensions and solvable Lie groups.

Roman Sauer: Spectral distribution of Laplace operators on amenable groups
I´ll present a formula computing the spectral distribution function (near zero) of the Laplacian on a finitely generated amenable group in terms of the (L2-)isoperimetric profile or the Folner function. This is joint work with Alexander Bendikov and Christophe Pittet.

Thomas Schick: The Atiyah conjecture about values of L2-Betti numbers
L2-Betti numbers are invariants of non-compact manifolds with a discrete cocompact group of symmetries. They have geometric but also algebraic meaning.
One of the important open questions asks about their possible values (for different types of discrete groups); with connections e.g. to Kaplanski´s zero-divisor conjecture for group rings. This range is predicted by the so-called Atiyah conjecutre.
We will report on several developments around this question, partly brand-new, partly somewhat older. In particular
- we will describe very precise counterexample to the Atiyah conjecture for groups with a lot of torsion elements, strengthening and refining Tim Austin´s results (which he will report on at another talk). This is joint work with Austin, Pichot, Zuk.
- we will describe very ring-theoretic methods which complete the discussion of the Atiyah conjecture for many groups with not much torsion (e.g. finite extensions of braid groups or right-angled Artin groups); this is joint work with Knebusch, Linnell.

Katrin Tent: Some new simple groups
Simple groups of Lie type play an important role in many areas of mathematics. We show to what extent they can be characterized by their actions and how to construct new simple groups with similar properties using methods from model theory and descriptive set theory.

Andreas Thom: Stability of unitary representations
I will report about joint work with Marc Burger and Narutaka Ozawa. We showed that for every epsilon, there exists delta, such that finite dimensional delta-unitary representation of certain lattices in semi-simple algebraic groups are epsilon-close to unitary representations. The important news is that delta can be choosen independent of the dimension (see also the talk by Narutaka Ozawa for more information). The proof relies on recent advances on bounded generation of certain lattices in semi-simple algebraic groups.

Karen Vogtmann: Rigidity phenomena in automorphism groups of free groups
Although the group of outer automorphisms of a free group of rank at least 3 is not a lattice in any semisimple Lie group, it behaves like one in many respects. In joint work with Martin Bridson we examine aspects of rigidity for these groups. For example, homomorphisms between lattices are constrained by the fact that they must extend to the ambient Lie group. We show that there are also strong constraints on maps between outer automorphism groups of free groups.