Farrell-Jones Conjecture Research Articles

L‐theory of C∗$C^*$‐algebras

AbstractWe establish a formula for the L‐theory spectrum of real ‐algebras from which we deduce a presentation of the L‐groups in terms of the topological K‐groups, extending all previously known results of this kind. Along the way, we extend the integral comparison map obtained in previous work by the first two authors to real ‐algebras and interpret it using topological Grothendieck–Witt theory. Finally, we use our results to give an integral comparison between the Baum–Connes conjecture and the L‐theoretic Farrell–Jones conjecture, and discuss our comparison map in terms of the signature operator on oriented manifolds.

An improvement of the Farrell-Jones conjecture for localising invariants

The Farrell-Jones conjecture for lax monoidal finitary localising invariants was recently proved by Bunke-Kasprowski-Winges [On the Farrell-Jones conjecture for localising invariants, 2021, arXiv:2111.02490]. In this short note, making use of the theory of noncommutative motives, we prove that the lax monoidal assumption is not necessary.

The Farrell–Jones Conjecture for Hyperbolic-by-Cyclic Groups

Abstract We prove the Farrell–Jones conjecture (FJC) for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the FJC by Bartels–Lück–Reich, as well as the structure theory of mapping tori by Dahmani–Krishna.

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

Abstract 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 let A Γ A_{\Gamma} be its associated Artin group; our new proof implies that if A T A_{T} is poly-free (resp. normally poly-free) for every clique 𝑇 in Γ, then A Γ A_{\Gamma} 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 A_{\Gamma} is an even Artin group such that each clique in Γ either has at most three vertices, has all of its labels at least 6, or is the join of these two types of cliques (the edges connecting the cliques are all labelled by 2), then A Γ A_{\Gamma} satisfies the Farrell–Jones Conjecture. In addition, our methods enable us to obtain results for general Artin groups.

TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

AbstractLet$\pi$be a group satisfying the Farrell–Jones conjecture and assume that$B\pi$is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group$\pi$whose canonical map to$B\pi$has degree 1, and show that two such manifolds ares-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If$\pi$is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

Pseudo-Sylvester domains and skew Laurent polynomials over firs

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with free-by-{infinite cyclic} and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell–Jones conjecture for normally poly-free groups and extends previous results of Linnell–Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.

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.

Finitely $\mathcal{F}$-amenable actions and decomposition complexity of groups

In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely \mathcal F -amenable" group action, where \mathcal F is a family of subgroups. We show how a finitely \mathcal F -amenable action of a countable group G on a compact metric space, where the asymptotic dimensions of the elements of \mathcal F are bounded from above, gives an upper bound for the asymptotic dimension of G viewed as a metric space with a proper left invariant metric. We generalize this to families \mathcal F whose elements are contained in a collection, \mathcal C , of metric families that satisfies some basic permanence properties: If G is a countable group and each element of \mathcal F belongs to \mathcal C and there exists a finitely \mathcal F -amenable action of G on a compact metrizable space, then G is in \mathcal C . Examples of such collections of metric families include: metric families with weak finite decomposition complexity, exact metric families, and metric families that coarsely embed into Hilbert space.

Acylindrical actions on trees and the Farrell–Jones conjecture

We show that for groups acting acylindrically on simplicial trees the K - and L -theoretic Farrell–Jones conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on their Bass-Serre trees we obtain an identification of the associatedWaldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalises a result by Lafont and Ortiz. For a regular ring and a strictly acylindrical action these Nil-groups vanish. In particular, all our results apply to amalgamated free products over malnormal subgroups.

On the stable Cannon Conjecture

The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable version that for any closed manifold N of dimension greater or equal to 2 there exists a closed manifold M together with a simple homotopy equivalence from M to the cartesian product of N and BG. If N is aspherical and pi_1(N) satisfies the Farrell-Jones Conjecture, then M is unique up to homeomorphism.

On the Farrell–Jones conjecture for algebraic K-theory of spaces : the Farrell–Hsiang method

We prove the Farrell-Jones Conjecture for algebraic K-theory of spaces for virtually poly-Z-groups. For this, we transfer the 'Farrell-Hsiang method' from the linear case to categories of equivariant, controlled retractive spaces.

The Farrell–Jones Conjecture for mapping class groups

We prove the Farrell–Jones Conjecture for mapping class groups. The proof uses the Masur–Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmuller space. The proof is presented in an axiomatic setup, extending the projection axioms in Bestvina et al. (Publ Math Inst Hautes Etudes Sci 122:1–64, 2015). More specifically, we prove that the action of $${{\,\mathrm{Mod(\Sigma )}\,}}$$ on the Thurston compactification of Teichmuller space is finitely $$\mathcal F$$ -amenable for the family $${\mathcal {F}}$$ consisting of virtual point stabilizers.

Farrell–Jones via Dehn fillings

Following the approach of Dahmani, Guirardel and Osin, we extend the group theoretical Dehn filling theorem to show that the pre-images of infinite order subgroups have a certain structure of a free product. We then apply this result to establish the Farrell–Jones conjecture for groups hyperbolic relative to a family of residually finite subgroups satisfying the Farrell–Jones conjecture, partially recovering a result of Bartels.

On the Farrell–Jones conjecture for Waldhausen’s A–theory

We prove the Farrell-Jones Conjecture for (non-connective) $A$-theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudo-isotopies in the topological, piecewise linear and smooth category.

Topological rigidity for FJ by the infinite cyclic group

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

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.

Long and thin covers for flow spaces

Long and thin covers of flow spaces are important ingredients in the proof of the Farrell–Jones conjecture for certain classes of groups, like hyperbolic and CAT(0)-groups. In this paper we provide an alternative construction of such covers which holds in a more general setting and simplifies some of the arguments.

On the relation between K- and L-theory of $$C^*$$ C ∗ -algebras

We prove the existence of a map of spectra $$\tau _A:kA \rightarrow \ell A$$ between connective topological K-theory and connective algebraic L-theory of a complex $$C^*$$ -algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence $$KA\left[ \tfrac{1}{2}\right] \xrightarrow {\simeq } LA\left[ \tfrac{1}{2}\right] $$ of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real $$C^*$$ -algebras. Using this we give a comparison between the real Baum–Connes conjecture and the L-theoretic Farrell–Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.

The Farrell–Jones conjecture for hyperbolic and CAT(0)-groups revisited

We generalize the proof of the Farrell–Jones conjecture for CAT(0)-groups to a larger class of groups, for example also containing all groups that act properly and cocompactly on a finite product of hyperbolic graphs. In particular, this gives a unified proof of the Farrell–Jones conjecture for CAT(0)- and hyperbolic groups.

Approximately fibering a manifold over an aspherical one

The paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers, i.e., is homotopic to manifold approximate fibration. We define obstructions in algebraic K-theory. Their vanishing is necessary and under certain conditions sufficient. Basic ingredients are Quinn’s thin h-cobordism theorem and end theorem, and knowledge about the Farrell–Jones conjectures in algebraic K- and L-theory and the MAF-rigidity conjecture by Hughes–Taylor–Williams.