Coarse Baum Connes Research Articles

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.

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.

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.

Expanders are counterexamples to the $\ell^p$ coarse Baum–Connes conjecture

We consider an \ell^p coarse Baum–Connes assembly map for 1 < p < \infty , and show that it is not surjective for expanders arising from residually finite hyperbolic 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.

$L^p$ 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.

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.

Geometric property (T) for non-discrete spaces

Geometric property (T) was defined by Willett and Yu, first for sequences of graphs and later for more general discrete spaces. Increasing sequences of graphs with geometric property (T) are expanders, and they are examples of coarse spaces for which the maximal coarse Baum-Connes assembly map fails to be surjective. Here, we give a broader definition of bounded geometry for coarse spaces, which includes non-discrete spaces. We define a generalisation of geometric property (T) for this class of spaces and show that it is a coarse invariant. Additionally, we characterise it in terms of spectral properties of Laplacians. We investigate geometric property (T) for manifolds and warped systems.

Coarse Baum-Connes conjecture and rigidity for Roe algebras

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if X and Y are two uniformly locally finite metric spaces such that their Roe algebras are ⁎-isomorphic, then X and Y are coarsely equivalent provided either X or Y satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but which does not coarsely embed into a Hilbert space.

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Herve Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then C*max(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture.

A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture

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.

On coarse geometric aspects of Hilbert geometry

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum–Connes conjecture holds for any two-dimensional Hilbert geometry via Yu’s theorem.

Controlled K-theory for groupoids & applications to Coarse Geometry

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 Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum–Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.

Kazhdan projections, random walks and ergodic theorems

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

The equivariant coarse Baum–Connes conjecture for spaces which admit an equivariant coarse embedding into Hilbert space

In this paper, we proved that the equivariant coarse Baum–Connes conjecture holds for a metric space with bounded geometry which admits an equivariant coarse embedding into Hilbert space.

The coarse Baum–Connes conjecture for Busemann nonpositively curved spaces

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.

The coarse Baum–Connes conjecture and related topics

The coarse Baum–Connes conjecture and related topics

Random graphs, weak coarse embeddings, and higher index theory

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum–Connes assembly map is injective; the coarse Baum–Connes assembly map is not surjective; the maximal coarse Baum–Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion.The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.