# Schedule of the Follow-up Workshop to TP Rigidity

All talks were 50 minutes with 10 minutes for discussion.

## Monday, April 27

10:30 - 11:00 |
Welcome coffee |

11:00 - 12:00 |
Wolfgang Lück: Introduction to the Farrell-Jones Conjecture (part I) |

12:00 - 14:00 |
Lunch break |

14:00 - 15:00 |
Discussion, talk or free time |

15:00 - 16:00 |
Wolfgang Lück: Introduction to the Farrell-Jones Conjecture (part II) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Arthur Bartels: Ways to prove instances of the Farrell-Jones Conjecture (part I) |

afterwards |
Reception |

## Tuesday, April 28

9:30 - 10:30 |
Romain Tessera: Finite decomposition complexity and the bounded Borel conjecture |

10:30 - 11:00 |
Coffee break, group photo |

11:00 - 12:00 |
Denis Osin: Acylindrically hyperbolic groups (part I) |

12:00 - 14:00 |
Lunch break |

14:00 - 15:00 |
Discussion, talk or free time |

15:00 - 16:00 |
Arthur Bartels: Ways to prove instances of the Farrell-Jones Conjecture (part II) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Discussion, talk or free time |

## Wednesday, April 29

9:30 - 10:30 |
Nigel Higson: Real reductive groups, K-theory and the Oka principle |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Guoliang Yu: Dimension, complexity and K-theory |

12:00 - 14:00 |
Lunch break |

14:00 - 14:25 |
Christian Wegner: The Farrell-Jones Conjecture for solvable groups |

14:30 - 14:55 |
Erik Pedersen: Bounded K- and L-theory |

15:00 - 16:00 |
Arthur Bartels: Ways to prove instances of the Farrell-Jones Conjecture (part III) |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Discussion, talk or free time |

18:30 - |
Dinner |

## Thursday, April 30

9:30 - 10:30 |
Martin Bridson: Fixed-point theorems and rigidity |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Denis Osin: Acylindrically hyperbolic groups (part II) |

12:00 - 14:00 |
Lunch break |

14:00 - 14:25 |
Wolfgang Lück: Transfer |

14:30 - 14:55 |
Goulnara Arzhantseva: "Constructing" monsters (groups) |

15:00 - 16:00 |
Mladen Bestvina: Hyperbolicity in mapping class groups and Out(F_{n}) |

16:00 - 16:30 |
Tea and cake |

16:30 - 16:55 |
Nigel Higson: Parabolic induction |

## Friday, May 1

9:30 - 10:30 |
Mark Sapir: Aspherical Higman embeddings |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Denis Osin: Acylindrically hyperbolic groups (part III) |

12:00 - |
Lunch break, end of workshop |

# Abstracts

## Arthur Bartels: Ways to prove instances of the Farrell-Jones Conjecture

Farrell-Jones used the geodesic flow to prove what is now known as their conjecture in many cases. I will survey some of the subsequent developments. In particular, I plan to discuss the axiomatizations of proof of the Farrell-Jones Conjecture. One point of view is that these proofs all fool the classifying space for the family of virtually cyclic groups into thinking that it is compact.

## Mladen Bestvina: Hyperbolicity in mapping class groups and Out(F_{n})

I will survey what is known about actions of mapping class groups and Out(F_{n}) on hyperbolic spaces.

## Martin Bridson: Fixed-point theorems and rigidity

In this talk I will discuss fixed-point theorems for group actions on CAT(0) and acyclic spaces, and related rigidity results.

## Nigel Higson: Real reductive groups, K-theory and the Oka principle

This talk will be about real reductive groups rather than discrete groups, so it will stand apart from the main themes of the conference, and will be about representations of groups that are projective (in the sense of module theory) rather than irreducible, so it will also stand apart from the main themes in reductive groups. But there will be close connections to the Baum-Connes isomorphism, and through that to issues of concern in both geometry and representation theory. I want to examine projective modules as families of irreducible modules. Whereas there is common agreement about the concept of irreducible module in representation theory, more than one viable notion of family is available. There is a good parallel with the theory of vector bundles, where one can consider holomorphic vector bundles, if the base is a complex manifold, or smooth vector bundles. The Oka principle reconciles the two when the base is a Stein manifold, and a parallel result for reductive groups would go a long way toward explaining the Baum-Connes isomorphism, through a connection to the Langlands classification theorem. I'll describe these things, along with the prospects for carrying over some of these ideas to other, more geometric, contexts.

## Wolfgang Lück: Introduction to the Farrell-Jones Conjecture (part I)

We start with a short discussion about algebraic K- and L-theory of group rings and their relevance. Then we give an introduction to the Farrell-Jones Conjecture in the special case, where the group G is torsionfree and the coefficient ring is regular. We discuss its main applications and give a survey about its status.

## Wolfgang Lück: Introduction to the Farrell-Jones Conjecture (part II)

We present the Farrell-Jones Conjecture in its general form. This requires some input about classifying spaces of families, equivariant homology theories and assembly maps. We explain some inheritance properties and give a first survey about the methods of proof focusing on the approach using geometric group theory.

## Denis Osin: Acylindrically hyperbolic groups

I will survey the recent progress in the study of groups acting acylindrically on a hyperbolic space. We will start with the basic theory and then discuss more advanced results and open questions.

## Mark Sapir: Aspherical Higman embeddings

I will explain how to embed every recursively presented aspherical group into a finitely presented group with a 2-dimensional K(.,1).

## Romain Tessera: Finite decomposition complexity and the bounded Borel conjecture

A metric space has finite decomposition complexity (FDC) if it can be broken into bounded pieces in a controlled way. This property have the following applications: if two aspherical manifolds M and N are such that their fundamental groups have FDC, then any quasi-isometry between their universal covers is at bounded distance from a homeomorphism. The manifold M also satisfies a weak version of the Borel conjecture. Moreover, the class of groups with FDC contains all hyperbolic groups, all linear groups and all elementary amenable groups.

## Christian Wegner: The Farrell-Jones Conjecture for solvable groups

I will give a sketch of the proof of the Farrell-Jones Conjecture for solvable groups.

## Guoliang Yu: Dimension, complexity and K-theory

In this talk, I will introduce a concept of dynamic asymptotic dimension and discuss its application to computation of K-theory. These computations will lead to various versions of integral Novikov conjectures. If time allows, I will discuss a notion of dynamic complexity, inspired by the geometric complexity introduced by Erik Guentner, Romain Tessera and myself. This is joint work with Erik Guentner and Rufus Willett.