Daniel Juan-Pineda, Stratos Prassidis
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.
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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 prove the A-theoretic Isomorphism Conjecture with coefficients and finite wreath products for solvable 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.
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 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.
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.
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 paper, we prove the K- and L-theoretical Isomorphism Conjecture for Baumslag-Solitar groups with coefficients in an additive category.
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.
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.
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.
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…
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].
We prove the $K$ and $L$ theoretic versions of the Fibered Isomorphism Conjecture of F. T. Farrell and L. E. Jones for braid groups on a surface.
Algebraic kk-theory and the KH-isomorphism conjecture
本研究借助同伦理论与伴随定理，建立带Weibel同伦K-理论系数的Davis-Lück同调与等变代数kk-理论的联系，并用等变代数kk-群刻画KH同构猜想装配映射的左侧，推进相关代数K-理论猜想研究。
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.
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…
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.
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, close…
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。
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.
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 $\mathbb{Z}$.
Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms
本文修改最终同调同构定义，引入n-伴随型奇异等价，发展相关理论并应用于同调猜想简化与性质保持，同时建立其与三角函子的对应关系，纳入3-伴随型奇异等价框架。
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…
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.
The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason's descent theorems.
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…
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…
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…
Noncommutative Quillen-Lichtenbaum Conjecture
本文通过非交换几何视角，扩展了经典Quillen-利文萨姆猜想至有限型分离复方案，并建立了代数与拓扑K群比较映射的同构范围。
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…
Every hyperbolic group acts continuously on its Gromov boundary. One can form the corresponding cross-product C*-algebra A. We show that there always exists a canonical Poincare duality map from the K-theory of A to the K-homology of A. We show that this map is an isomorphism when the group in question is the free group on two generators. There is a direct connection between our constructions and the Baum-Connes Conjecture, and we use the latter to deduce our result.
We give a new proof of some cases of the Baum-Connes conjecture along the lines of a proof of the Farrell-Jones conjecture.
We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is the first step to prove the algebraic K-theory isomorphism conjecture for simplicial rings. We construct a category of controlled simplicial modules, show that it has the structure of a Waldhausen category and discuss its algebraic K-theory. We lay emphasis on detailed proofs. Highlights include the discussion of a simplicial cylinder functor, the gluing lem…
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.
Leavitt path algebras, $B_\infty$-algebras and Keller's conjecture for singular Hochschild cohomology
针对无汇点有限箭图，本文通过建立$B_\infty$-代数同伦范畴中的复形同构，验证了根平方零代数的Keller猜想，证明其在单点（余）扩张等变换下不变，结论覆盖所有有限维gentle代数。
We develop a generalization of quantitative $K$-theory, which we call controlled $K$-theory. It is powerful enough to study the $K$-theory of crossed product of $C^*$-algebras by action of étale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse …
Controlled objects in left-exact $\infty$-categories and the Novikov conjecture
本研究构造了带G作用的左正合∞-范畴的等变X受控对象范畴，结合代数K-理论得到新等变粗同调理论，将相关装配映射单射性结果应用于左正合∞-范畴的代数K-理论。
On the Connes-Kasparov isomorphism, II: The Vogan classification of essential components in the tempered dual
本文作为系列论文第二篇，利用David Vogan对缓增对偶的描述计算Connes-Kasparov态射，从而完成实约化群约化C*-代数的Morita等价分类并验证Connes-Kasparov猜想。
We consider the equivariant K-theory of a real semisimple Lie group which acts on the (complex) flag variety of its complexification group. We construct an assemble map in the framework of KK-theory and then we prove that it is an isomorphism. The prove relies on a careful study of the orbits of the real group action on the flag variety and then piecing together different orbits. This result is a special case of the Baum-Connes conjecture with coefficients.
On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual
本文是两篇系列论文的第一篇，计算连通线性实约化群的约化C*-代数在Morita等价意义下的结构，并在缓增表示论结果基础上计算Connes-Kasparov态射，从而验证Connes-Kasparov猜想。
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