Baum-Connes Conjecture Research Articles

Permanence of the torsion-freeness property for divisible discrete quantum subgroups

We prove that torsion-freeness in the sense of Meyer-Nest is preserved under divisible discrete quantum subgroups. As a consequence, we obtain some stability results of the torsion-freeness property for relevant constructions of quantum groups (quantum (semi-)direct products, compact bicrossed products and quantum free products). We improve some stability results concerning the Baum-Connes conjecture appearing already in a previous work of the author. For instance, we show that the (resp. strong) Baum-Connes conjecture is preserved by discrete quantum subgroups (without any torsion-freeness or divisibility assumption).

Equivariant K-Homology and K-Theory for Some Discrete Planar Affine Groups

Abstract We consider the semi-direct products $G={\mathbb{Z}}^{2}\rtimes GL_{2}({\mathbb{Z}}), {\mathbb{Z}}^{2}\rtimes SL_{2}({\mathbb{Z}})$, and ${\mathbb{Z}}^{2}\rtimes \Gamma (2)$ (where $\Gamma (2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum–Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^{*}$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the first and third groups, the computations in $K_{0}$ provide explicit generators that are matched by the Baum–Connes assembly map.

A CATEGORICAL APPROACH TO THE BAUM–CONNES CONJECTURE FOR ÉTALE GROUPOIDS

AbstractWe consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid $C^*$ -algebras under homotopy of twists.

The equivariant coarse Baum–Connes conjecture for actions by a-T-menable groups

The equivariant coarse Baum–Connes conjecture for actions by a-T-menable groups

The equivariant coarse Baum–Connes conjecture for metric spaces with proper group actions

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 \Gamma acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/\Gamma admits a coarse embedding into Hilbert space and \Gamma is amenable, and that the \Gamma -orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum–Connes conjecture holds for (X,\Gamma) . Along the way, we prove a K -theoretic amenability statement for the \Gamma -space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K -theory.

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.

Deformation of algebras associated with group cocycles

We study deformation of algebras with coaction symmetry of reduced algebras of discrete groups, where the deformation parameter is givenby a continuous family of group 2 -cocycles. When the group satisfies the Baum–Connes conjecture with coefficients, we obtain an isomorphism of K-groups of the deformed algebras. This extends both the \theta -deformation of Rieffel on \mathbb{T}^n -actions, and a recent result of Echterhoff, Lück, Phillips, and Walters on the K-groups on the twisted group algebras.

K-Theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations

Let [Formula: see text] be a discrete metric space with bounded geometry. In this paper, we show that if [Formula: see text] admits an “A-by-CE” coarse fibration, then the canonical quotient map [Formula: see text] from the maximal Roe algebra to the Roe algebra of [Formula: see text], and the canonical quotient map [Formula: see text] from the maximal uniform Roe algebra to the uniform Roe algebra of [Formula: see text], induce isomorphisms on [Formula: see text]-theory. A typical example of such a space arises from a sequence of group extensions [Formula: see text] such that the sequence [Formula: see text] has Yu’s property A, and the sequence [Formula: see text] admits a coarse embedding into Hilbert space. This extends an early result of Špakula and Willett [Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013) 35–68] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum–Connes conjecture holds for a large class of metric spaces which may not admit a fibered coarse embedding into Hilbert space.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and \widetilde{M} is a \Gamma -regular covering space of M . Let \langle \alpha \rangle be the conjugacy class of a non-identity element \alpha\in \Gamma . Suppose \{\Gamma_i\} is a sequence of finite-index normal subgroups of \Gamma that distinguishes \langle \alpha \rangle . Let \pi_{\Gamma_i} be the quotient map from \Gamma to \Gamma/\Gamma_i and \langle \pi_{\Gamma_i}(\alpha) \rangle the conjugacy class of \pi_{\Gamma_i}(\alpha) in \Gamma/\Gamma_i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of \widetilde{M} at the conjugacy class \langle \alpha \rangle is equal to the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle , where M_{\Gamma_i}= \widetilde{M}/\Gamma_i is the finite-sheeted covering space of M determined by \Gamma_i . In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle converges, under the assumption that the rational maximal Baum–Connes conjecture holds for \Gamma .

The coarse Baum–Connes conjecture for certain relative expanders

Let (1→Nm→Gm→Qm→1)m∈N be a sequence of extensions of finite groups such that their coarse disjoint unions have bounded geometry. In this paper, we show that if the coarse disjoint unions of (Nm)m∈N and (Qm)m∈N are coarsely embeddable into Hilbert space, then the coarse Baum–Connes conjecture holds for the coarse disjoint union of (Gm)m∈N.As an application, the coarse Baum–Connes conjecture holds for the relative expanders constructed by G. Arzhantseva and R. Tessera, and the special box spaces of free groups discovered by T. Delabie and A. Khukhro, which do not coarsely embed into Hilbert space, yet do not contain a weakly embedded expander. This enlarges the class of metric spaces known to satisfy the coarse Baum–Connes conjecture. In particular, it solves an open problem raised by G. Arzhantseva and R. Tessera on the coarse Baum–Connes conjecture for relative expanders.

K-theory of noncommutative Bernoulli shifts

For a large class of C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras A, we calculate the K-theory of reduced crossed products A⊗G⋊rG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{\\otimes G}\\rtimes _rG$$\\end{document} of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras of wreath products H≀G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\wr G$$\\end{document} for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.

XXL type Artin groups are CAT(0) and acylindrically hyperbolic

We describe a simple locally CAT(0) classifying space for XXL type Artin groups (with all labels at least 5). Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that XXL type Artin groups are acylindrically hyperbolic. Together with Property RD proved by Ciobanu, Holt and Rees, the CAT(0) property implies the Baum–Connes conjecture for all XXL type Artin groups.

Measured Asymptotic Expanders and Rigidity for Roe Algebras

Abstract In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity of Roe algebras. Consequently, we obtain the rigidity for all bounded geometry spaces that coarsely embed into some $L^p$-space for $p\in [1,\infty )$. Moreover, we also verify rigidity for the box spaces constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not coarsely embed into any $L^p$-space. The key step in our proof of rigidity is showing that a block-rank-one (ghost) projection on a sparse space $X$ belongs to the Roe algebra $C^{\ast }(X)$ if and only if $X$ consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum–Connes conjecture.

Equivariant Callias index theory via coarse geometry

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the $K$-theory of a group $C^*$-algebra.

Cubulation of some triangle-free Artin groups

We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly cubulated, even virtually. The existence of such a proper action has many interesting consequences for the group, notably the Haagerup property, and the Baum–Connes conjecture with coefficients.

Cardinal-indexed classifying spaces for families of subgroups of any topological group

For G a topological group, existence theorems by Milnor (1956), Gelfand–Fuks (1968), and Segal (1975) of classifying spaces for principal G-bundles are generalized to G-spaces with torsion. Namely, any G-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For G a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact G. Namely, the G-homeomorphism types of proper G-spaces over a metric space correspond to stratified-homotopy classes of orbit classifying maps.The former existence result is enabled by Segal's clever but esoteric use of non-Hausdorff spaces. The latter uniqueness result is enabled by our own development of equivariant ANR theory for noncompact Lie G. Applications include the existence part of classification for unstructured fiber bundles with locally compact Hausdorff fiber and with locally connected base or fiber, as well as for equivariant principal bundles which in certain cases via other models is due to Lashof–May (1986) and to Lück–Uribe (2014). From a categorical perspective, our general model EFκG is a final object inspired by the formulation of the Baum–Connes conjecture (1994).

$L^p$ coarse Baum–Connes conjecture and $K$-theory for $L^p$ Roe algebras

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 is independent of p\in(1,\infty) for spaces with finite asymptotic dimension.

Hecke operators in Bredon (co)homology, K-(co)homology and Bianchi groups

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-theory of the reduced [Formula: see text]-algebra of the group. We show the power of this method giving explicit computations for the group [Formula: see text]. In order to carry out these computations we use an Atiyah–Segal type spectral sequence together with the Bredon homology of the classifying space for proper actions.

On the structure of asymptotic expanders

In this paper, we study several analytic and graph-theoretic properties of asymptotic expanders. We show that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we use asymptotic expanders to provide uncountably many new counterexamples to the coarse Baum–Connes conjecture. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C⁎-algebraic characterisation of expanders for vertex-transitive graphs. We achieve these results by showing that a sequence of asymptotic expanders always admits a “uniform exhaustion by expanders”. This also implies that asymptotic expanders cannot be coarsely embedded into any Lp-space.