Thomas Schick
In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups, or certain fundamental groups of complements of links in S^3.
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Bourbaki seminar 1062, October 2012.
Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).
On the Baum-Connes conjecture for $D_{\infty}$
本研究遵循Higson与Kasparov的思路，对无限二面体群的Baum-Connes猜想的证明进行了阐述。
The Baum-Connes conjecture for extensions
本文通过反例证明群扩展中巴乌斯-康纳斯猜想的永久性假设无法简化。
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.
We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting.
We show that the image of Connes-Karoubi-Chern character, restricted to the image of the Baum-Connes assembly map in the Bott-periodized topological K-theory of the complex group algebra, lies in the elliptic summand of the (periodic) cyclic homology of the group algebra. This implies that for any (weighted) ell-1 completion of the group algebra, rational surjectivity of the Baum-Connes assembly map implies the generalized Bass conjecture for that algebra.
We document some versions, in real K-theory, of well-known properties of the coarse assembly map in complex K-theory. These results are well-known, but difficult to find in the literature.
The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group $Γ$ acts properly and isometrically on a discrete metric space $X$ with bounded geometry, not necessarily cocompact. We show that if the quotient space $X/Γ$ admits a coarse embedding into Hilbert space and $Γ$ is amenable, and that the $Γ$-orbits in $X$ are uniformly equivariantly coarsely equiva…
The main purpose of this paper is to modify the orbit method for the Baum-Connes conjecture as developed by Chabert, Echterhoff and Nest in their proof of the Connes-Kasparov conjecture for almost connected groups \cite{MR2010742} in order to deal with linear algebraic groups over local function fields (i.e., non-archimedean local fields of positive characteristic). As a consequence, we verify the Baum-Connes conjecture for certain Levi-decomposable linear algebraic groups over local function fields. One of these is the Jacobi group, which is the semidirect product of the symplectic group and …
We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.
We prove the Baum--Connes conjecture with arbitrary coefficients for some classes of groups: (1) Linear algebraic groups over a non-archimedean local field. (2) Linear algebraic groups over the adeles of a global field k, provided that at every archimedean place of k the associated Lie group is amenable. (3) All closed subgroups of the above groups. This includes linear algebraic groups over global fields - with the same condition as in (2). Update: proof of main results incomplete, maybe not repairable.
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 prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.
In this paper, we prove the K- and L-theoretical Isomorphism Conjecture for Baumslag-Solitar groups with coefficients in an additive category.
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 define and compare two bivariant generalizations of the topological $K$-group $K^\top(G)$ for a topological group $G$. We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.
On the quantitative coarse Baum-Connes conjecture with coefficients
本文提出了针对真度量空间的带系数定量粗Baum-Connes猜想（QCBC），证明其可由带系数的粗Baum-Connes猜想导出，并指出QCBC最终可归结为一列有界度量空间的一致定量猜想。
We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. Especially, the group satisfies the analytic Novikov conjecture.
We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field.
The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason's descent theorems.
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.
In this article we give a characterisation of the Baum-Connes assembly map with coefficients. The technical tools needed are the K-theory of C*-categories, and equivariant KK-theory in the world of groupoids.
Given a (not necessarily discrete) proper metric space $M$ with bounded geometry, we define a groupoid $G(M)$. We show that the coarse Baum--Connes conjecture with coefficients, which states that the assembly map with coefficients for G(M) is an isomorphism, is hereditary by taking closed subspaces.
The coarse Baum-Connes conjecture with filtered coefficients and product metric spaces
本文受定量K-理论启发，提出带滤过系数的粗Baum-Connes猜想，推广原始猜想，并证明其满足类在积下封闭，进而发现新的积度量空间例子。
We introduce a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. The method can simplify and clarify proofs of some of the known cases of the conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, a similar idea will be used to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space.
We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $ SU_q(2) $ of Woronowicz. As an illustration of this result we determine the $ K $-groups of quantum automorphism groups of simple matrix algebras.
Coronae of product spaces and the Coarse Baum-Connes conjecture
本文研究乘积空间与乘积群的coarse Baum-Connes猜想，通过构造并分析proper度量空间乘积的紧化及其边界corona，证明了满足特定假设的CAT(0)群、多项循环群与相对双曲群的乘积满足该猜想。
We establish a coarse version of the Cartan-Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum-Connes conjecture. Combined with the result of Osajda-Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum-Connes conjecture.
In this paper, we verify the $L^p$ coarse Baum-Connes conjecture for spaces with finite asymptotic dimension for $p\in[1,\infty)$. We also show that the $K$-theory of $L^p$ Roe algebras are independent of $p\in(1,\infty)$ for spaces with finite asymptotic dimension.
We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particular, we give a geometric proof of this conjecture for spaces of graphs that have large girth and bounded vertex degree. We then connect the boundary conjecture to the coarse Baum-Connes conjecture using homological methods, which allows us to exhibit all the current uniformly discr…
Let $G$ denote a countable inverse semigroup. We construct a kind of a Baum--Connes map $K(\tilde A \rtimes G) \rightarrow K(A \rtimes G)$ by a categorial approach via localization of triangulated categories, developed by R. Meyer and R. Nest for groups $G$. We allow the coefficient algebras $A$ to be in a special class of algebras called fibered $G$-algebras. This note continues and fixes our preprint "Attempts to define a Baum--Connes map via localization of categories for inverse semigroups".
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
$\ell^p$-coarse Baum-Connes conjecture for $\ell^{q}$-coarse embeddable spaces
证明了可粗嵌入$\ell^q$空间的空间满足$\ell^p$粗Baum-Connes猜想，覆盖所有$p,q\in[1,\infty)$。
We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated 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 $Γ$.
We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoid…
Alternative titles of this paper would have been `Index theory without index' or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C*-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl-Moyal quantization on manifolds, C*-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum-Connes conjecture for smooth groupoids as described by Connes in hi…
This is a survey on Kasparov's bivariant $KK$-theory in connection with the Baum-Connes conjecture on the $K$-theory of crossed products $A\rtimes_rG$ by actions of a locally compact group $G$ on a C*-algebra $A$. In particular we shall discuss Kasparov's Dirac dual-Dirac method as well as the permanence properties of the conjecture and the "Going-Down principle" for the left hand side of the conjecture, which often allows to reduce $K$-theory computations for $A\rtimes_rG$ to computations for crossed products by compact subgroups of $G$. We give several applications for this principle includi…
The quantitative coarse Baum-Connes conjecture for free products
本文在自由积G*H中证明了定量粗 Baum-Connes 猜想，前提是该猜想在G和H上成立。
Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the Twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in [1] or by Tu,Xu and Laurent-Gengoux in [24]. These Twisted geometric K-homology groups are the left hand sides of the twisted geometric Baum-Connes assembly maps recently constructed in [9] and hence one can transfer the multiplicative structure via the Baum-Connes map to the Twisted K-the…
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.
In this note we set a configuration space description of the equivariant connective K-homology groups with coefficients in a unital C*-algebra for proper actions. Over this model we define a connective assembly map and prove that in this setting is possible to recover the analytic assembly map.
Coarse geometry of Hecke pairs and the Baum-Connes conjecture
本研究借助Hecke对陪集空间与Schlichting完备化的粗几何，证明了余Haagerup性质下Baum-Connes和Novikov猜想的新稳定性结果，推广了既有结论并给出满足带系数Baum-Connes猜想的S-算术群新实例。
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.
We prove that if the classical Baum-Connes conjecture in complex K-theory is true (for a given discrete group G), then the conjecture is also true in the real case (for the same group G). The essential ingredients of the proof are the descent theorem in topological K-theory (math.KT/0509396) and a Paschke duality for C*-algebras proved by John Roe (Q.J. Math. 55 (2004) 325-331). According to S. Stolz, this result is linked with the existence of a Riemannian metric of positive scalar curvature on a compact connected spin manifold with G as fundamental group.
We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ γ$-element and that $ γ= 1 $. It follows that free orthogonal quantum groups are $ K $-amenable. We compute explicitly their $ K $-theory and deduce in the unimodular case that the corresponding reduced $ C^* $-algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum-Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equival…
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 introduce the notion of proper Kasparov cycles for Kasparov's G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the no…
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 introduce a notion of fibred coarse embedding into Hilbert space for metric spaces, which is a generalization of Gromov's notion of coarse embedding into Hilbert space. It turns out that a large class of expander graphs admit such an embedding. We show that the maximal coarse Baum-Connes conjecture holds for metric spaces with bounded geometry which admit a fibred coarse embedding into Hilbert space.
Meyer and Nest showed that the Baum--Connes map is equivalent to a map on $K$-theory of two different crossed products. This approach is strongly categorial in method since its bases is to regard Kasparov's theory $KK^G$ as a triangulated category. We have tried to translate this approach to the realm of inverse semigroup equivariant $C^*$-algebras but can prove the existence of a Baum--Connes map only under some unverified additional assumptions which we however strongly motivate. Some of our results may be of independent interest, for example Bott periodicity, the definition of induction fun…
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between $\ell_2$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.
$L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry
本文通过 $C_0$ 粗结构研究 $L^p$ 粗 Baum-Contes 猜想，证明有限维simplicial复形在统一球面度量下成立，并构造障碍群，证明有限渐近维数下障碍群消失，从而提供新的证明。
Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture
本文通过构造来自剩余有限双曲群的膨胀图，证明了 $\ell^p$ 粗 Baum-Connes 装配映射在 $1<p<\infty$ 时不是满射，从而给出该猜想的反例。
We describe a geometric counterpart of the Baum-Connes map for the p-adic group GL(n).
The twisted coarse Baum--Connes conjecture and relative hyperbolic groups
本文引入稳定粗代数并提出扭曲粗Baum--Connes猜想，证明其在粗等价等操作下保持，并应用于相对双曲群，得出群满足猜想当且仅当其每个双曲子群满足。
In this work, the continuously controlled techniques developed by Carlsson and Pedersen are used to prove that the Baum-Connes map is a split injection for groups satisfying certain geometric conditions.
Using the unbounded picture of analytical K-homology, we associate a well-defined K-homology class to an unbounded symmetric operator satisfying certain mild technical conditions. We also establish an ``addition formula'' for the Dirac operator on the circle and for the Dolbeault operator on closed surfaces. Two proofs are provided, one using topology and the other one, surprisingly involved, sticking to analysis, on the basis of the previous result. As a second application, we construct, in a purely analytical language, various homomorphisms linking the homology of a group in low degree, the …
We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett extends to correspondences when r…
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