The K-theoretic Farrell-Jones Conjecture for hyperbolic groups

Coefficients for the Farrell-Jones Conjecture

We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version…

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 $d$, $|d |>1$.

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.

The A-theoretic Farrell-Jones Conjecture for virtually solvable groups

We prove the A-theoretic Farrell-Jones Conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S-arithmetic groups and lattices in almost connected Lie groups.

On the Farrell-Jones and related Conjectures

These extended notes are based on a series of six lectures presented at the summer school ``Cohomology of groups and algebraic $K$-theory'' which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the Farrell-Jones and the Baum-Connes Conjecture.

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.

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.

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.

Farrell-Jones猜想对局部化不变量的一个改进

An improvement of the Farrell-Jones conjecture for localising invariants

本文利用非交换动机理论，改进了Farrell-Jones猜想，证明其对于局部化不变量无需lax monoidal假设。

On the Farrell-Jones Conjecture and its applications

We present the status of the Farrell-Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true and study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably.

关于局部化不变量的Farrell-Jones猜想

On the Farrell-Jones conjecture for localising invariants

本研究证明了有限$\u0002$-顺从群及更一般的Dress-Farrell-Hsiang-Jones群的、系数在左正合$\u221e$-范畴中的Farrell-Jones猜想，统一了加法范畴K-理论等相关论证，并将其推广至$\u2164_1$-环谱上的完美模范畴。

Model theoretic reformulation of the Baum-Connes and Farrell-Jones conjectures

The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason's descent theorems.

The Farrell-Jones conjecture for S-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.

A Dirac Morphism for the Farrell-Jones Isomorphism Conjecture in K-Theory

We construct a Dirac morphism and prove that if this Dirac morphism is invertible, then the isomorphism conjecture for non-connective algebraic K-theory holds true.

On the Farrell-Jones Conjecture for higher algebraic K-theory

We prove the Farrell-Jones Isomorphism Conjecture about the algebraic K-theory of a group ring RG in the case where the group G is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring R is an arbitrary associative ring with unit and the result applies to all dimensions.

全不连通群的 Farrell-Jones 猜想的继承性质

Inheritance properties of the Farrell-Jones Conjecture for totally disconnected groups

研究全不连通群 Hecke 代数的 K 理论 Farrell-Jones 猜想，证明其对闭子群具有继承性，并为后续约化 p-adic 群情形建立构造与工具。

A remark on the Farrell-Jones conjecture

Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring $R$ and a thick subcategory $\mathsf{C}$ of perfect $R$-complexes such that the Waldhausen $K$-theory space $\mathrm{K}(\mathsf{C})$ is equivalent to a rational Eilenberg-Maclane space.

On passage to over-groups of finite indices of the Farrell-Jones conjecture

We use the controlled algebra approach to study the problem that whether the Farrell-Jones conjecture is closed under passage to over-groups of finite indices. Our study shows that this problem is closely related to a general problem in algebraic $K$- and $L$-theories. We use induction theory to study this general problem. This requires an extension of the classical induction theorem for $K$- and $L$- theories of finite groups with coefficients in rings to with twisted coefficients in additive categories. This extension is well-known to experts, but a detailed proof does not exist in the liter…

The Farrell-Jones Conjecture for normally poly-free groups

We prove the $K$- and $L$-theoretic Farrell-Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A\rtimes \mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell--Jones Conjecture for free-by-cyclic groups.

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

We show that contracting self-similar groups satisfy the Farrell-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.

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.

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.

相对超几何群的自同构与Farrell-Jones猜想

Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture

本文证明了相对超几何群的悬垂群和单端超几何群的悬垂群在A-、K-和L-理论中满足Farrell-Jones猜想，并探讨了其在自同构群和延伸中的应用。

Relating the Farrell Nil-groups to the Waldhausen Nil-groups

We prove that the Waldhausen Nil-group associated to a virtually cyclic groups that surjects onto the infinite dihedral group vanishes if and only if the corresponding Farrell Nil-group associated to the canonical index two subgroup is trivial. The proof uses the transfer map to establish one direction, and uses controlled pseudo-isotopy techniques of Farrell-Jones to establish the reverse implication.

The Farrell-Jones isomorphism conjecture for 3-manifold groups

We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove here, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to pr…

Poly-freeness of Artin groups and the Farrell-Jones Conjecture

We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let $Γ$ be a finite simplicial graph with all edges labelled by positive even integers and $A_Γ$ be its associated Artin group; our new proof implies that if $A_T$ is poly-free (resp. normally poly-free) for every clique $T$ in $Γ$, then $A_Γ$ is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell--Jones Conjecture for even Artin groups. In particular, we show that if $A_Γ$ i…

Topological Rigidity for FJ by the Infinite Cyclic Group

We call a group FJ if it satisfies the $K$- and $L$-theoretic Farrell-Jones conjecture with coefficients in $\mathbb Z$. We show that if $G$ is FJ, then the simple Borel conjecture (in dimensions $\ge 5$) holds for every group of the form $G\rtimes\mathbb Z$. If in addition $Wh(G\times \mathbb Z)=0$, which is true for all known torsion free FJ groups, then the bordism Borel conjecture (in dimensions $n\ge 5$) holds for $G\rtimes\mathbb Z$. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion free group $G$ satisfies the $L$-theoretic…

The Bounded Isomorphism Conjecture for Spaces of Graphs with Large Girth

In this article we study a coarse version of the K-theoretic Farrell-Jones conjecture we call coarse or bounded isomorphism conjecture. With techniques that have already been used to prove the Farrell-Jones conjecture for hyperbolic groups we are able to verify the bounded isomorphism conjecture for spaces of graphs with large girth and bounded geometry.

Trees are 1-Transfer

The K-theoretic Farrell-Jones isomorphism conjecture for a group ring $R[G]$ has been proved for several groups. The toolbox for proving the Farrell-Jones conjecture for a given group depends on some geometric properties of the group as it is the case of hyperbolic groups. The technique used to prove it for hyperbolic groups $G$ relies in the concept of an $N$-transfer space endowed with a $G$ action. In this work, we give an explicit construction of a $1$-transfer space.

The Bounded Isomorphism Conjecture for Box Spaces of Residually Finite Groups

In this article we study a coarse version of the $K$-theoretic Farrell--Jones conjecture we call coarse or bounded isomorphism conjecture. Using controlled category theory we are able to translate this conjecture for asymptotically faithful covers into a more familiar form. This allows us to prove the conjecture for box spaces of residually finite groups whose Farrell--Jones assembly map with coefficients is an isomorphism.

Isomorphism Conjecture for homotopy K-theory and groups acting on trees

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory.

On crossed product rings with twisted involutions, their module categories and L-theory

We study the Farrell-Jones Conjecture with coefficients in an additive G-category with involution. This is a variant of the L-theoretic Farrell-Jones Conjecture which originally deals with group rings with the standard involution. We show that this formulation of the conjecture can be applied to crossed product rings R*G equipped with twisted involutions and automatically implies the a priori more general fibered version.

L-theory of groups with unstable derived series

In this short note we prove that the Farrell-Jones Fibered Isomorphism Conjecture in L-theory, after inverting 2, is true for a group whose some derived subgroup is free.

Inheritance of Isomorphism Conjectures under colimits

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.

The isomorphism conjecture for Artin groups

We prove the Farrell-Jones fibered isomorphism conjecture for several classes of Artin groups of finite and affine types. As a consequence, we compute explicitly the surgery obstruction groups of the finite type pure Artin groups.

K-theory and actions on Euclidean retracts

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class groups.

Algebraic K-theory over the infinite dihedral group: a controlled topology approach

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group which maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell-Bass Nil groups of the index two subgroup which maps surjectively to the infinite cycli…

Algebraic K-theory over the infinite dihedral group: an algebraic approach

We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.

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).

Algebraic K-theory of mapping class groups

We prove that the Fibered Isomorphism Conjecture of T. Farrell and L. Jones holds for various mapping class groups. In many cases, we explicitly calculate the lower algebraic K-groups, showing that they do not always vanish.

约化p-adic群的代数K-理论

Algebraic K-theory of reductive p-adic groups

本文提出并证明了约化p-adic群Hecke代数的K-理论的C-op-Farrell-Jones猜想，从而可计算其射影类群，并给出负K-理论消失及高K-群与Bruhat-Tits建筑同调的联系。

Asymptotic transfer maps in parametrized K-theory

We define asymptotic transfers in bounded K-theory together with a context where this can be done in great generality. Controlled algebra plays a central role in many advances in geometric topology, including recent work on Novikov, Borel, and Farrell-Jones conjectures. One of the features that appears in various manifestations throughout the subject, starting with the original work of Farrell and Jones, is an asymptotic transfer whose meaning and construction depend on the geometric circumstances. We first develop a general framework that allows us to construct a version of asymptotic transfe…

The Baum-Connes conjecture, Swan group actions and controlled topology

We give a new proof of some cases of the Baum-Connes conjecture along the lines of a proof of the Farrell-Jones conjecture.

The isomorphism conjecture in L-theory: graphs of groups

We study the Fibered Isomorphism Conjecture of Farrell and Jones in L-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in B of Theorem 1.1 we prove the L-theory version of [[7], Theorem 1.2].

On the vanishing of the lower K-theory of the Holomorph of a free group on two generators

We show that the holomorph of the free group on two generators satisfies the Farrell-Jones Fibered Isomorphism Conjecture. As a consequence, we show that the lower K-theory of the above group vanishes.

On Farrell-Tate cohomology of SL\_2 over S-integers

In this paper, we provide number-theoretic formulas for Farrell-Tate cohomology for SL\_2 over rings of S-integers in number fields satisfying a weak regularity assumption. These formulas describe group cohomology above the virtual cohomological dimension, and can be used to study some questions in homology of linear groups. We expose three applications, to (I) detection questions for the Quillen conjecture,(II) the existence of transfers for the Friedlander--Milnor conjecture,(III) cohomology of SL\_2 over number fields.

On Farrell-Tate cohomology of GL(3) over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary q…

Assembly Maps for Group Extensions in $K$-Theory and $L$-Theory with Twisted Coefficients

In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in algebraic $K$-theory and $L$-theory with twisted coefficients.

Algebraic K-theory of groups wreath product with finite groups

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example for discrete subgroups of Lie groups, virtually poly-infinite cyclic groups, Artin braid groups, a class of virtually poly-surface groups and virtually solvable linear group. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G with H for H a finite group. We also prove the conjecture for some other classes of groups.

Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods

We give a concise introduction to the Farrell-Jones Conjecture in algebraic $K$-theory and to some of its applications. We survey the current status of the conjecture, and we illustrate the two main tools that are used to attack it: controlled algebra and trace methods.

Artin 群的特定结构与同构猜想

A certain structure of Artin groups and the isomorphism conjecture

本文发现一大类 Artin 群的归纳结构，并据此推导出有限实、复和仿射类型 Artin 群的 Farrell-Jones 同构猜想。

Topological K-homology for non-proper actions and assembly maps

For a countable discrete group $G$, we construct a new and concrete model for the equivariant topological $K$-homology theory of $G$, which is defined for all $G$-actions, not just for proper $G$-actions. The construction of our model combines some of the ideas from the localization algebra approach to the coarse Baum-Connes conjecture and the controlled algebra approach to the Farrell-Jones conjecture. It brings new ideas into the study of the Baum-Connes conjecture. First, as the model is defined for all $G$-actions, we are able to define relative assembly maps. We prove, by using an inducti…

The isomorphism conjecture in L-theory

This is the first of three articles on the Fibered Isomorphism Conjecture of Farrell and Jones for L-theory. We apply the general techniques developed in [15] and [16] to the L-theory case of the conjecture and prove several results. Here we prove the conjecture, after inverting 2, for poly-free groups. In particular, it follows for braid groups. We also prove the conjecture for some classes of groups without inverting 2. In fact we consider a general class of groups satisfying certain conditions which includes the above groups and some other important classes of groups. We check that the prop…

Bredon Cohomology, K theory and K homology of Pullbacks of groups

We develop an Eilenberg-Moore spectral sequence to compute Bredon cohomology of spaces with an action of a group given as a pullback. Using several other spectral sequences, and positive results on the Baum-Connes Conjecture, we are able to compute Equivariant K-theory and K-Homology of the reduced group C*-algebra of a 6-dimensional crystallographic group $Γ$ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones Conjecture to give a vanishing result for the negative algebraic K-theory of the integral group ring of $Γ$.