HIM Trimester Program on
Rigidity
Workshop: Rigidity in cohomology, Ktheory, 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).
Schedule

Monday, November 23

Tuesday, November 24

Wednesday, November 25

Thursday, November 26

Friday, November 27
Abstracts
Tim Austin: A counterexample to a conjecture of Atiyah
A 1972 conjecture of Atiyah asserts that for any finitelygenerated
group, the L^2 Betti numbers of any cocompact free proper \Gmanifold 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 nonpositively curved groups
In this talk the connection between topological rigidity of aspherical
manifolds and nonpositively 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
nonlinear 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.
PierreEmmanuel Caprace: A lattice in a product of more than two KacMoody groups is arithmetic
Margulis´ seminal works imply that irreducible lattices in semisimple linear groups of rank at least two are (superrigid and) arithmetic.
BurgerMozes lattices in the automorphism group of a product of two trees, and KacMoody lattices in the automorphism group of a product of two buildings, provide examples of (superrigid) nonarithmetic lattices beyond the linear world. One expects however that the existence of nonarithmetic 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 KacMoody 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 online resource for
information about manifolds being hosted at the HIM: http://www.manifoldatlas.him.unibonn.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
Ktheory of BG and the Ktheory of the reduced C*algebra of G. For the
latter we use the BaumConnes conjecture.
On the geometric side, we prove the GromovLawsonRosenberg 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
FarrellJones Conjecture and is the first classification of manifolds whose
fundamental group is not built from torsionfree 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 2dimensional 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 measuretheoretic 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 nonzero degree.
In this talk, we will concentrate on the question of which manifolds are not dominated by a nontrivial
product of manifolds. For rationally essential manifolds, we give a sufficient condition for nonpresentability
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 torsionfree 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 (n1)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 superrigidity theorem for products.
Narutaka Ozawa: Quasihomomorphism 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 quasihomomorphisms with noncommutative targets and εrepresentations.
Irine Peng: Quasiisometric rigidities in solvable groups
Building on the works of FarbMosher and EskinFisherWhyte,
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 L2Betti numbers
L2Betti numbers are invariants of noncompact 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 zerodivisor conjecture for group rings. This range is predicted by the socalled
Atiyah conjecutre.
We will report on several developments around this question, partly brandnew, 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 ringtheoretic methods which complete the discussion of the Atiyah conjecture for many groups with
not much torsion (e.g. finite extensions of braid groups or rightangled 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 deltaunitary representation of certain lattices in semisimple algebraic groups
are epsilonclose 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 semisimple 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.