# Schedule of the Workshop "The Farrell-Jones conjecture"

## Monday, October 24

10:30 - 11:00 |
Registration & Welcome coffee |

11:00 - 12:00 |
Dan Ramras: Coassembly for representation spaces |

12:00 - 14:45 |
Lunch break and free time |

14:45 - 15:45 |
Nigel Higson: Isomorphism conjectures for non-discrete groups |

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

16:30 - 17:30 |
Romain Tessera: Finite decomposition complexity: an introduction |

afterwards |
Reception |

## Tuesday, October 25

09:30 - 10:30 |
Alexander Engel: The Burghelea conjecture |

10:30 - 11:00 |
Group photo and coffee break |

11:00 - 12:00 |
Holger Reich: Assembly maps for algebraic K-theory and topological cyclic homology (part 1) |

12:00 - 14:45 |
Lunch break and free time |

14:45 - 15:45 |
Holger Reich: Assembly maps for algebraic K-theory and topological cyclic homology (part 2) |

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

16:30 - 17:30 |
Xiaolei Wu: On the finiteness of the classifying space for the family of virtually cyclic subgroups |

## Wednesday, October 26

09:30 - 10:30 |
Daniel Kasprowski: On the K-theory of groups with finite decomposition complexity |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Grigori Avramidi: Topology of ends of finite volume, non-positively curved manifolds |

12:00 - |
Lunch break, free afternoon |

18:30 - |
Conference dinner in the Restaurant Meyer's (Clemens-August-Str. 51a) |

## Thursday, October 27

09:30 - 10:30 |
Ulrich Bunke: A motivic view of coarse homotopy theory |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Arthur Bartels: The Farrell-Jones conjecture for mapping class groups (part 1) |

12:00 - 14:45 |
Lunch break and free time |

14:45 - 15:45 |
Arthur Bartels: The Farrell-Jones conjecture for mapping class groups (part 2) |

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

16:30 - 17:30 |
Henrik Rüping: The Farrell-Jones conjecture and bicombings |

## Friday, October 28

09:30 - 10:30 |
Christoph Winges: On the isomorphism conjecture for Waldhausen's algebraic K-theory of spaces |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Wolfgang Lück: The group cohomology of certain crystallographic groups and applications |

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

# Abstracts

## Grigori Avramidi: Topology of ends of finite volume, non-positively curved manifolds

The structure of ends of nonpositively curved, locally symmetric manifolds is very well understood. In this talk, I will explain features of the locally symmetric situation that are true for more general nonpositively curved manifolds. This is joint work with Tam Nguyen Phan.

## Arthur Bartels: The Farrell-Jones conjecture for mapping class groups

The main step in the proof of the Farrell-Jones conjecture for mapping class groups is the verification of a regularity condition, related to amenability, for the action of the mapping class group on the space of projective measured foliations. I will discuss axioms that allow the verification of this property. These axioms are on one hand concerned with Teichmüller flow and on the other hand concerned with subsurface projections.I will also discuss why these axioms are satisfied in the case of the mapping class group. This is joint work with Mladen Bestvina.

## Ulrich Bunke: A motivic view of coarse homotopy theory

In this talk I introduce the category of bornological coarse spaces as a general framework for coarse geometry. Then is discuss the axioms for coarse homology theories. Coarse motivic spectra a defined as the target of the universal coarse homology theory. I will explain how the known examples of coarse homology theory fit into this picture.

One of the goals of the project is to understand parts of the known proofs of the Farrell-Jones conjecture and versions of the coarse Baum-Connes conjecture as the verification of motivic statements. Consequently, these arguments apply not only to the homology theory considered in the respective case, but to any coarse homology theory.

## Alexander Engel: The Burghelea conjecture

In the first half of the talk we will quickly introduce the algebraic Baum-Connes conjecture and the Burghelea conjecture and discuss their relation. In the second half I will report about recent results on the Burghelea conjecture. This is joint work with Michal Marcinkowski.

## Nigel Higson: Isomorphism conjectures for non-discrete groups

I shall discuss aspects of the C*-algebraic version of the Farrell-Jones conjecture (namely the Baum-Connes conjecture) for Lie groups and p-adic groups. The conjecture has already been proved in the cases I shall discuss, but both sides of the isomorphism assume different characters from the discrete case in the light of representation theory, and many unsolved problems remain. There is at least the potential for an interesting interaction between geometric or topological methods and representation theory.

## Daniel Kasprowski: On the K-theory of groups with finite decomposition complexity

We will show that for every ring R the assembly map for the family of finite subgroups in algebraic K-theory is split injective for certain groups with finite decomposition complexity. In particular, this holds for discrete subgroups of connected Lie groups and for linear groups which admit a finite-dimensional model for the classifying space for proper actions.

## Wolfgang Lück: The group cohomology of certain crystallographic groups and applications

There is a conjecture due to Adem-Ge-Pan-Petroysan about the cohomology of certain crystallographic groups with cyclic holonomy. We will disprove it in general, but prove it in the case where the action of the holonomy group is free. We discuss applications to the classification of Cuntz-Li-C*-algebras associated to number fields, to the classification of total spaces of certain torus bundles over lens spaces and to the existence of Riemannian metrics with positive sectional curvature on a smooth manifold.

## Dan Ramras: Coassembly for representation spaces

I'll discuss models for a coassembly map (the topological Atiyah-Segal map) from representation spaces to topological K-theory. At its most basic, this map carries a linear representation of a discrete group G to the K-theory class of its associated vector bundle over BG. This map can be realized as a map of ring spectra, and I'll explain its relationships to the (strong) Novikov Conjecture, spaces of flat connections, and families of flat vector bundles. Additionally, I'll explain a version of this map that takes into account a family of subgroups of G, in a manner dual to the Farrell-Jones assembly map.

## Holger Reich: Assembly maps for algebraic K-theory and topological cyclic homology

This is a report on joint work with Wolfgang Lück (Bonn), John Rognes (Oslo), and Marco Varisco (Albany).

The first talk will explain how the cyclotomic trace map from algebraic K-theory to topological cyclic homology and Bökstedt-Hsiang-Madsen’s functor C can be used to prove rational injectivity results about the Farrell-Jones assembly map. The technique applies under mild homological finiteness conditions on the group, and produces in particular injectivity results for Whitehead groups. Results about higher K-theory need to assume a weak version of the Leopoldt-Schneider conjecture for cyclotomic fields.

The second talk will study assembly maps for topological cyclic homology of group algebras integrally. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism. For infinite groups, we establish pro-isomorphism, split injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, in the case of hyperbolic groups and of virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is split injective but in general not surjective.

## Henrik Rüping: The Farrell-Jones conjecture and bicombings

A. Bartels and W. Lück proved the Farrell-Jones conjecture for CAT(0)-groups. In this talk I will report on join work with D. Kasprowski where we weakened the assumption of being CAT(0) and I will give some examples of groups satisfying this weaker assumptions. For example those also hold for Gromov hyperbolic groups and groups acting geometrically on a product of Gromov hyperbolic spaces.

## Romain Tessera: Finite decomposition complexity: an introduction

We shall introduce FDC, explain why it is useful in surgery theory, and sketch a proof that linear groups satisfy this property.

## Christoph Winges: On the isomorphism conjecture for Waldhausen's algebraic K-theory of spaces

I will survey recent progress on the isomorphism conjecture for Waldhausen's "algebraic K-theory of spaces" functor, and how this relates to the original isomorphism conjecture of Farrell and Jones as well as the computation of homotopy groups of automorphism groups of closed, aspherical manifolds. This talk covers joint work with Enkelmann, Kasprowski, Lück, Pieper, Ullmann and Wegner.

## Xiaolei Wu: On the finiteness of the classifying space for the family of virtually cyclic subgroups

In this talk, I will discuss the classifying space for the family of virtually cyclic subgroups. In particular, I will introduce an interesting conjecture due to Juan-Pineda and Leary. The conjecture says a group admits a finite model for the classifying space for the family of virtually cyclic subgroups iff it is virtually cyclic. I will talk about some recent progress on this conjecture. This is a joint work with Tim von Puttkamer.