Farrell-Jones Conjecture Research Articles

Coarse flow spaces for relatively hyperbolic groups

We introduce coarse flow spaces for relatively hyperbolic groups and use them to verify a regularity condition for the action of relatively hyperbolic groups on their boundaries. As an application the Farrell–Jones conjecture for relatively hyperbolic groups can be reduced to the peripheral subgroups (up to index-2 overgroups in the $L$-theory case).

Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups

We prove the K- and the $L$-theoretic Farrell-Jones conjecture with coefficients in additive categories and with finite wreath products for arbitrary lattices in virtually connected Lie groups.

Farrell–Jones Conjecture for fundamental groups of graphs of virtually cyclic groups

In this note, we prove the K- and L-theoretic Farrell–Jones Conjecture with coefficients in an additive category for fundamental groups of graphs of virtually cyclic groups.

Singular coefficients in the K–theoretic Farrell–Jones conjecture

Let G be a group and k a field of characteristic zero. We prove that if the Farrell-Jones conjecture for the K-theory of R[G] is satisfied for every smooth k-algebra R, then it is also satisfied for every commutative k-algebra R.

Topological rigidity and actions on contractible manifolds with discrete singular set

The problem of equivariant rigidity is the Γ \Gamma -homeomorphism classification of Γ \Gamma -actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of Γ \Gamma . In other words, this is the classification of cocompact E f i n Γ E_{\mathrm {fin}} \Gamma -manifolds. We use surgery theory, algebraic K K -theory, and the Farrell–Jones Conjecture to give this classification for a family of groups which satisfy the property that the normalizers of nontrivial finite subgroups are themselves finite. More generally, we study cocompact proper actions of these groups on contractible manifolds and prove that the E f i n E_{\mathrm {fin}} condition is always satisfied.

The Farrell–Jones conjecture forS-arithmetic groups

This paper contains the results of my PhD-thesis. I will show the K- and L-theoretic Farrell-Jones conjecture (FJC) for the general linear groups over the rationals and over the rational functions over a finite field. This especially implies the conjecture for all S-arithmetic groups.

On the transfer reducibility of certain Farrell–Hsiang groups

We show how the existing proof of the Farrell-Jones Conjecture for virtually poly-$\mathbb{Z}$-groups can be improved to rely only on the usual inheritance properties in combination with transfer reducibility as a sufficient criterion for the validity of the conjecture.

The Farrell–Jones conjecture for fundamental groups of graphs of abelian groups

We show that the Farrell–Jones conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag–Solitar groups.

The Farrell–Jones conjecture for virtually solvable groups:

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for virtually solvable groups.

The Farrell–Jones conjecture for some nearly crystallographic groups

In this paper, we prove the K-theoretical and L-theoretical Farrell-Jones Conjecture with coefficients in an additive category for nearly crystallographic groups of the form $\mathbb{Q}^n \rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{Q}^n$ as an irreducible integer matrix with determinant $d$, $|d |>1$.

Compact operators and algebraic K-theory for groups which act properly and isometrically on Hilbert space

Abstract We prove the K-theoretic Farrell–Jones conjecture for groups with the Haagerup approximation property and coefficient rings and C * C^{*} -algebras which are stable with respect to compact operators. We use this and Higson–Kasparov’s result that the Baum–Connes conjecture holds for such a group G, to show that the algebraic and the C * C^{*} -crossed product of G with a stable separable G- C * C^{*} -algebra have the same K-theory.

The Farrell–Jones Conjecture for the solvable Baumslag–Solitar groups

In this paper, we prove the Farrell–Jones Conjecture for the solvable Baumslag–Solitar groups with coefficients in an additive category. We also extend our results to groups of the form, Z[1/p] semidirect product with any virtually cyclic group, where p is a prime number.

The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups

The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups

The Farrell–Jones conjecture for graph products

We show that the class of groups satisfying the K- and L-theoretic Farrell-Jones conjecture is closed under taking graph products of groups.

Topologie

The Oberwolfach conference “Topologie” is one of the few occasions where researchers from many different areas in algebraic and geometric topology are able to meet and exchange ideas. Accordingly, the program covered a wide range of new developments in such fields as classification of manifolds, isomorphism conjectures, geometric topology, and homotopy theory. More specifically, we discussed progress on problems such as the Farrell-Jones conjecture, higher dimensional analogues of Harer’s homological stability of automorphism groups of manifolds and new algebraic concepts for equivariant spectra, to mention just a few subjects. One of the highlights was a series of four talks on new methods and results about the Farrell-Jones conjecture by Arthur Bartels and Wolfgang Lück.

K- and L-theory of group rings over GL n (Z)

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL n (Z).

(Self-)similar groups and the Farrell–Jones conjectures

We show that contracting self-similar groups satisfy the Farrel–Jones conjectures as soon as their universal contracting cover is non-positively curved. This applies in particular to bounded self-similar groups. We define, along the way, a general notion of contraction for groups acting on a rooted tree in a not necessarily self-similar manner.

Homotopy invariance of 4-manifold decompositions: Connected sums

Given any homotopy equivalence f:M→X1#⋯#Xn of closed orientable 4-manifolds, where each fundamental group π1(Xi) satisfies Freedmanʼs Null Disc Lemma, we show that M is topologically h-cobordant to a connected sum M′=M1′#⋯#Mn′ such that f is h-bordant to some f1′#⋯#fn′ with each fi′:Mi′→Xi a homotopy equivalence. Moreover, such a replacement M′ of M is unique up to a connected sum of h-cobordisms. In summary, the existence and uniqueness, up to h-cobordism, of connected sum decompositions of such orientable 4-manifolds M is an invariant of homotopy equivalence.Also we establish that the Borel Conjecture is true in dimension 4, up to s-cobordism, if the fundamental group satisfies the Farrell–Jones Conjecture.

Geodesic flow for CAT(0)–groups

We associate to a CAT(0)-space a flow space that can be used as the replacement for the geodesic flow on the sphere tangent bundle of a Riemannian manifold. We use this flow space to prove that CAT(0)-group are transfer reducible over the family of virtually cyclic groups. This result is an important ingredient in our proof of the Farrell-Jones Conjecture for these groups.