1,721,071 research outputs found
The coarse Baum-Connes conjecture via Co coarse geometry
No full textThe 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
Finite asymptotic dimension for CAT(0) cube complexes
No full textWe prove that the asymptotic dimension of a finite-dimensional CAT(0) cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every CAT(0) cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups
Co coarse geometry and scalar curvature
No full textIn this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C*-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature ?(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces. <br/
Simultaneous metrizability of coarse spaces
No full textA metric space can be naturally endowed with both a topology and a coarse structure. We examine the converse to this. Given a topology and a coarse structure we give necessary and sufficient conditions for the existence of a metric giving rise to both of these. We conclude with an application to the construction of the coarse assembly map.<br/
Expanders and property A
No full textWe give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu’s property A
Spaces of graphs, boundary groupoids and the coarse Baum-Connes conjecture
No full textWe 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 discrete counterexamples to the coarse Baum–Connes conjecture in an elementary way
Pairings, duality, amenability and bounded cohomology
No full textWe give a new perspective on the homological characterisations of amenability given by Johnson and Ringrose in the context of bounded cohomology and by Block and Weinberger in the context of uniformly finite homology. We examine the interaction between their theories and explain the relationship between these characterisations. We apply these ideas to give a new proof of non- vanishing for the bounded cohomology of a free group
Building weight-free Følner sets for Yu's Property A in coarse geometry
No full textIn this note we study the natural question of when the generalised Følner sets exhibiting property A can be chosen to be subsets of the space itself. We show that for many property A spaces X, this is indeed possible. Specifically this holds: for all discrete bounded geometry metric spaces which coarsely have all components unbounded; for all countable discrete groups; and for all box spaces.<br/
A cohomological characterisation of Yu's Property A for metric spaces
No full textWe introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invariant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A
Stratified Langlands duality in the A<sub>n</sub> tower
No full textLet S k denote a maximal torus in the complex Lie group G=SL n (C)/C k and let T k denote a maximal torus in its compact real form SU n (C)/C k , where k divides n . Let W denote the Weyl group of G , namely the symmetric group S n . We elucidate the structure of the extended quotient S k //W as an algebraic variety and of T k //W as a topological space, in both cases describing them as bundles over unions of tori. Corresponding to the invariance of K -theory under Langlands duality, this calculation provides a homotopy equivalence between T k //W and its dual T nk //W . Hence there is an isomorphism in cohomology for the extended quotients which is stratified as a direct sum over conjugacy classes of the Weyl group. We use our formula to compute a number of examples
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