The coarse geometric Novikov conjecture for subspaces of non-positively curved manifolds

In this paper, we prove the coarse geometric Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature.     

On Banach spaces with Kasparov and Yu's Property (H)

A new geometric property of Banach spaces recently introduced by G. Kasparov and G. Yu, called Property (H), has important applications to the strong Novikov conjecture and the coarse Novikov conjecture, yet is far from being well understood. In this paper, we investigate various uniformly continuous maps on unit spheres of Banach spaces and prove that all separable Banach lattices with nontrivial cotype have Property (H).     

Property A and Uniform Embeddability of Metric Spaces Under Decompositions of Finite Depth

Property A and uniform embeddability are notions of metric geometry which imply the coarse Baum-Connes conjecture and the Novikov conjecture. In this paper, the authors prove the permanence properties of property A and uniform embeddability of metric spaces under large scale decompositions of finite depth.     

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.

     

Hyperbolic metric spaces and the exotic cohomology Novikov conjecture

A geometric version of the Novikov conjecture states that certain cohomology classes of a complete metric space arise from an ideal boundary. We prove this for spaces hyperbolic in the sense of Gromov.     

Metric sparsification and operator norm localization

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. In particular, we introduce a sufficient geometric condition (called metric sparsification) for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that a sequence of expanding graphs does not possess the operator norm localization property.     

Permanence of Metric Sparsification Property under Finite Decomposition Complexity

The notions of metric sparsification property and finite decomposition com- plexity are recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper, it is proved that a metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property. As a consequence, the authors obtain an alterna- tive proof of a very recent result by Guentner, Tessera and Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.     

Operator norm localization for linear groups and its application to K-theory

We prove the operator norm localization property for linear groups. As an application we prove the coarse Novikov conjecture for box spaces of a linear group.     

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Hervé 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.     

Operator norm localization property of metric spaces under finite decomposition complexity

The notions of operator norm localization property and finite decomposition complexity were recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper we show that a metric space X has weak finite decomposition complexity with respect to the operator norm localization property if and only if X itself has the operator norm localization property. It follows that any metric space with finite decomposition complexity has the operator norm localization property. In particular, we obtain an alternative way to prove a very recent result by E. Guentner, R. Tessera and G. Yu that all countable linear groups have the operator norm localization property.     

Non-uniformly Contractible Spaces and the Coarse Novikov Conjecture

In this paper bounded surgery obstruction groups for some noncoarsely uniformly contractible spaces are computed. These spaces are constructed out of the union of countably many spheres of fixed dimension and increasing radii. These results support the coarse Novikov conjecture: the coarse homology of the space with coefficients in theory injects into the bounded L-group.     

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov Conjecture on higher signatures in the context of the algebraic geometry of (nonsingular) complex projective varieties. This conjecture asserts that certain ``higher Todd genera' are birational invariants. This implies birational invariance of certain extra combinations of Chern classes (beyond just the classical Todd genus) in the case of varieties with large fundamental group (in the topological sense). We prove the conjecture under the assumption of the ``strong Novikov Conjecture' for the fundamental group, which is known to be correct for many groups of geometric interest. We also show that, in a certain sense, our conjecture is best possible.

     

Constructions preserving Hilbert space uniform embeddability of discrete groups

Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

     

Fibred coarse embeddings,a-T-menability and the coarse analogue of the Novikov conjecture

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 property that is sufficient for the maximal analogue to the coarse Baum–Connes conjecture and in this paper we connect this property to the traditional coarse Baum–Connes conjecture using a restriction of the coarse groupoid and homological algebra. Additionally we use this results to give a characterisation of the a-T-menability for residually finite discrete groups.     

Higher index theory for certain expanders and Gromov monster groups,II

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum–Connes assembly map is an isomorphism for the associated metric space X. As discussed in the first paper in this series, this has applications to the Baum–Connes conjecture for ‘Gromov monster’ groups.We also introduce a new property, ‘geometric property (T)’. For the metric space associated to a sequence of graphs, this property is an obstruction to the maximal coarse assembly map being an isomorphism. This enables us to distinguish between expanders with girth tending to infinity, and, for example, those constructed from property (T) groups.     

An etale approach to the Novikov conjecture

We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactification of an EΓ. We then show that for groups of finite asymptotic dimension, the Higson compactification is mod p acyclic for all p and deduce the integral Novikov conjecture for these groups. © 2007 Wiley Periodicals, Inc.     

The maximal coarse Baum–Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space

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.     

The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF^∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF^∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes-Moscovici and Dru?u-Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF^∞ and isocohomological.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

C*-Algebras and Controlled Topology

We describe some aspects of the relationship between the controlled topology and C*-algebra approaches to the Novikov conjecture.