On the isomorphism conjecture for groups acting on trees
S. K. Roushon
备注：
`The Fibered Isomorphism Conjecture wreath product with finite groups' was introduced and its general properties were studied in the initial versions itself of this paper, in arXiv:math/0601736 and in arXiv:math/0405211
We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These conditions are satisfied in several cases of the conjecture. We prove some general results on the conjecture for the pseudoisotopy theory for groups acting on trees with residually finite vertex stabilizers. In particular, we study situations when the stabilizers belong to the following classes of groups: polycyclic groups, finitely generated nilpotent groups, closed surface groups, finitely generated abelian groups and virtually cyclic groups.
Finally, we provide explicit examples of groups for which we have proved the conjecture in this article and show that these groups were not considered before. Furthermore, we deduce that these groups are neither hyperbolic nor CAT(0).
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.
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.
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].
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…
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.
We use a hocolim approach to the Isomorphism Conjecture in K-Theory to analyze the case of groups of the form $G\rtimes Z$ and $G_1*_{G}G_2$. As an important corollary we prove that the isomorphism conjecture in K-Theory holds for a finitely generated free group.
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.
The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RG, where G is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and an arbitrary coefficient ring R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RG in terms of the K-theory of R and the homology of the group.
Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture for K-theoretic ones.
In this article we prove that the $KH$-asembly map, as defined by Bartels and L{ü}ck, can be described in terms of the algebraic $KK$-theory of Cortinas and Thom. The $KK$-theory description of the $KH$-assembly map is similar to that of the Baum-Connes assembly map. In very elementary cases, methods used to prove the Baum-Connes conjecture also apply to the $KH$-isomorphism conjecture.
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.
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.
In this paper we discuss how the question about the rationality of L^2-Betti numbers is related to the Isomorphism Conjecture in algebraic K-theory and why in this context noncommutative localization appears as an important tool.
In this article we study the K- and L-theory of groups acting on trees. We consider the problem in the context of the fibered isomorphism conjecture of Farrell and Jones. We show that in the class of residually finite groups it is enough to prove the conjecture for finitely presented groups with one end. Also, we deduce that the conjecture is true for the fundamental groups of graphs of finite groups and of trees of virtually cyclic groups. To motivate the reader we include a survey on some classical works on this subject.
We show that for various natural classes of groups and appropriately defined K- and L-theoretic functors, injectivity or bijectivity of the assembly map follows from the Isomorphism Conjecture being true for acyclic groups lying within that class.
We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K_0 classifies them completely. For arbitrary Leavitt path algebras, it is conjectured that K^{\gr}_0 classifies them completely \cite{hazann}. We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple ones) if their K^{\gr}_0-groups of their Leavitt path algebras are isomorphic then their K_0-groups are isomorphic as well.
The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C^*-algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with infinite cyclic kernel or to the family of finite cyclic subgroups. Roughly speaking, we ext…
This article has two purposes. In \cite{R3} (math.KT/0405211) we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by \cal C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of \cal C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M in {\cal C} there is a closed 3-manifold P su…
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…
The Bost conjecture with C*-algebra coefficients for locally compact Hausdorff groups passes to open subgroups. We also prove that if a locally compact Hausdorff group acts on a tree, then the Bost conjecture with C*-coefficients is true for the group if and only if it is true for the stabilisers of the vertices.
Based on results by S.K. Roushon (math.KT/0408243 and math.KT/0405211) this thesis summarizes in an axiomatic way when a Meta-Isomorphism-Conjecture in the sense of Lueck and Reich (math.KT/0402405) is true for fundamental groups of 3-dimensional manifolds. In particular we prove that the fibered Farrell-Jones isomorphism conjectures for L-theory and algebraic K-theory are true for this class of groups if they are true for semidirect products of $\mathbb{Z^2}$ with $\mathbb{Z}$.
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 associated Waldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalizes a result by Lafont and Ortiz. For a regular ring and a strictly acylindrical a…
Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
We give a new proof of the Baum--Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg-Valette complex of a CAT(0)-cubical space introduced by the first three authors, and the direct splitting method in Kasparov theory developed by the last author.
We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.
In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - t…
Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevi…
We prove that Friedlander's generalized isomorphism conjecture on the cohomology of algebraic groups, and hence the Isomorphism Conjecture for the cohomology of the complex algebraic Lie group G(C) made discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F), where F is the algebraic closure of a finite field.
Let $G$ be a group and let $E$ be a functor from small $\Z$-linear categories to spectra. Also let $A$ be a ring with a $G$-action. Under mild conditions on $E$ and $A$ one can define an equivariant homology theory of $G$-simplicial sets $H^G(-,E(A))$ with the property that if $H\subset G$ is a subgroup, then \[ H^G_*(G/H,E(A))=E_*(A\rtimes H) \] If now $\cF$ is a nonempty family of subgroups of $G$, closed under conjugation and under subgroups, then there is a model category structure on $G$-simplicial sets such that a map $X\to Y$ is a weak equivalence (resp. a fibration) if and only if $X^H…
Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an…
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.
We present a geometric approach to the Baum-Connes conjecture with coefficients for Gromov monster groups via a theorem of Khoskham and Skandalis. Secondly, we use recent results concerning the a-T-menability at infinity of large girth expanders to exhibit a family of coefficients for a Gromov monster group for which the Baum-Connes conjecture is an isomorphism.
This is a study of algebras with involution that become isomorphic over a separable closure of the base field to a tensor product of two composition algebras. We classify these algebras, provide criteria for isomorphism and isotopy, and determine their automorphism groups, structure groups, and norm-similitude groups. The most interesting cases of these algebras are the $64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2 \times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ act faithfully by automorphisms and norm-preserving isotopies, respectively. We c…
