Erik Guentner, Jerome Kaminker
We study the connection between the condition that the reduced C*-algebra of a finitely presented group is exact and the Novikov conjecture holding. The main result states that if the group is strongly exact in the sense that the inclusion of the group C*-algebra into the uniform Roe algebra of the group is a nuclear embedding then the Novikov conjecture holds for that group.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Strong relative Novikov conjecture for coarsely embeddable groups
本文证明了粗可嵌入Hilbert空间的任意群对满足强相对Novikov猜想，推进了群论与几何拓扑中的相关猜想研究。
The aim of this paper is to introduce an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig using the Atiyah-Singer families index theorem, and partly by Carlsson's deformation $K$--theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the …
In this article we formulate a version of the analytic Novikov conjecture for semigroups rather than groups, and show that the descent argument from coarse geometry generalises effectively to this new situation.
The coarse geometric Novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero. The purpose of this paper is to prove the coarse geometric Novikov conjecture for spaces which admit a (coarse) uniform embedding into a uniformly convex Banach space.
The Coarse Novikov Conjecture for Finite Products of Fibred Coarsely Embeddable Spaces
本文证明了有界几何真度量空间的有限乘积满足粗Novikov猜想，通过构造粗真代数和多阶段Roe代数及指标映射，建立了连接迭代相对指标理论与全局猜想的约化定理。
Quantitative index, Novikov conjecture and coarse decomposability
本文定义考虑伪微分传播的定量装配映射估计，建立其与 Novikov 猜想及粗分解的联系，并几何证明有限分解复杂性群的 Novikov 猜想。
$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture
本文研究了任意群对的相对Baum-Connes组装映射，并在群满足特定几何条件时，应用该方法成功解决了相对Novikov猜想。
The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group $Γ$ acts on a bounded geometric space $X$ properly, isometrically, and with bounded distortion, $X/Γ$ and $Γ$ admit coarse embeddings into Hilbert space, then the $Γ$-equivariant coarse Novikov conjecture holds for $X$. Here bounded distortion means that for any $γ\in…
We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .
In this paper, we introduce a strongly quasi-local version of the coarse Novikov conjecture, which states that certain assembly map from the coarse $K$-homology of a metric space to the $K$-theory of its strongly quasi-local algebra is injective. We prove that the conjecture holds for metric spaces with bounded geometry which can be coarsely embedded into Banach spaces with Property (H), introduced by Kasparov and Yu. Besides, we also generalise the notion of strong quasi-locality to proper metric spaces and provide a (strongly) quasi-local picture for $K$-homology.
Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$ when the normal subgroup $N$ and the quotient group $G/N$ are coarsely embeddable into Hilbert spaces. As a result, the group $G$ satisfies the Novikov conjecture under the same hypothesis on $N$ and $G/N$.
We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra holds for many other exotic group C*-algebras, in particular the one associated to the smallest strongly Morita compatible and exact crossed product functor used in the new version of the Baum-Connes conjecture. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.
We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite\-/dimensional symmetric spaces. Thus our main theorem can be consid…
The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces
本研究通过算子K-理论与群在无穷维非正曲率空间连续场上的作用方法，证明了紧光滑流形微分同胚群离散子群的Novikov猜想，移除了此前研究的保体积条件。
In this paper, we prove the algebraic K-theory Novikov conjecture for group algebras over the ring of Schatten class operators. The main technical tool in the proof is an explicit construction of the Connes-Chern character.
We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L2-de Rham cohomology theory satisfying Poincare duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mishchenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov c…
Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture
本文研究可等变粗嵌入到可容许 Hilbert-Hadamard 空间的度量空间的等变粗 Novikov 猜想，先证明无挠群情形下等变粗强 Novikov 猜想有理成立，再引入有理解析等变粗 Novikov 猜想并去掉无挠假设，证明相应装配映射为有理单射。
We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficie…
Let $(1\to N_n\to G_n\to Q_n \to 1)_{n\in \mathbb{N}}$ be a sequence of extensions of countable discrete groups. Endow $(G_n)_{n\in \mathbb{N}}$ with metrics associated to proper length functions on $(G_n)_{n\in \mathbb{N}}$ respectively such that the sequence of metric spaces $(G_n)_{n\in \mathbb{N}}$ have uniform bounded geometry. We show that if $(N_n)_{n\in \mathbb{N}}$ and $(Q_n)_{n\in \mathbb{N}}$ are coarsely embeddable into Hilbert space, then the coarse Novikov conjecture holds for the sequence $(G_n)_{n\in \mathbb{N}}$, which may not admit a coarse embedding into Hilbert space.
We formulate and prove a Bott periodicity theorem for an $\ell^p$-space ($1\leq p<\infty$). For a proper metric space $X$ with bounded geometry, we introduce a version of $K$-homology at infinity, denoted by $K_*^{\infty}(X)$, and the Roe algebra at infinity, denoted by $C^*_{\infty}(X)$. Then the coarse assembly map descents to a map from $\lim_{d\to\infty}K_*^{\infty}(P_d(X))$ to $K_*(C^*_{\infty}(X))$, called the coarse assembly map at infinity. We show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show…
The connection between the coarse geometry of metric spaces and analytic properties of topological groupoids is well known. One of the main results of Skandalis, Tu and Yu is that a space admits a coarse embedding into Hilbert space if and only if a certain associated topological groupoid is a-T-menable. This groupoid characterisation then reduces the proof that the coarse Baum-Connes conjecture holds for a coarsely embeddable space to known results for a-T-menable groupoids. The property of admitting a fibred coarse embedding into Hilbert space was introduced by Chen, Wang and Yu to provide a…
The Baum-Connes and the Mishchenko-Kasparov assembly maps for group extensions
本文研究了带系数的巴乌-康纳斯装配映射e*和米斯钦科-卡斯帕罗夫装配映射μ*在群扩张中的单射性、满射性和同构性，证明了在特定条件下这些映射的性质可以传递到扩展群。
In this paper, we introduce and study the persistent approximation property for quantitative K-theory of filtered C*-algebras. In the case of crossed product C*-algebras, the persistent approximation property follows from the Baum-Connes conjecture with coefficients. We also discuss some applications of the quantitative K-theory to the Novikov conjecture.
We construct a Baum--Connes assembly map localised at the unit element of a discrete group $Γ$. This morphism, called $μ_τ$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[τ]\in KK_{\mathbb{R}}^Γ(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $Γ$. We show that the corresponding $τ$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $μ_τ$ is functorial with respect to the group $Γ$.
\noindent Let $A$ be an arbitrary $C^*$ algebra. In \cite{BH} Blackadar and Handelman conjectured the set of lower semicontinuous dimension functions on $A$ to be pointwise dense in the set $DF(A)$ of all dimension functions on $A$ and $DF(A)$ to be a Choquet simplex. We provide an equivalent condition for the first conjecture for unital $A$. Then by applying this condition we confirm the first Conjecture for all unital $A$ for which either the radius of comparison is finite or the semigroup $W(A)$ is almost unperforated. As far as we know the most general results on the first Conjecture up to…
Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.
In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^\ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum-Connes conjecture holds for …
An exotic crossed product is a way of associating a C*-algebra to each C*-dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C*-algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness. In this paper, we survey some constructions of exotic group algebras and exotic crossed products…
Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity
本文利用Roe代数商的相对高阶指标理论，研究非紧流形无穷远处正数量曲率度量的存在性障碍，提出相对粗Baum-Connes猜想并验证其部分情形，应用于扩展图乘积。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。