1,720,966 research outputs found
Uniform K-homology theory
No full textWe define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C?-algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C?-algebras. We show that our theory has a Mayer–Vietoris sequence. We prove that for a torsion-free countable discrete group ?, the direct limit of the uniform K-homology of the Rips complexes of ?, is isomorphic to the left-hand side of the Baum–Connes conjecture with coefficients in ???. In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras
Almost homomorphisms of compact groups
No full textA continuous mapping between compact topological groups which is "almost" a homomorphism need not be uniformly close to a homomorphism. The distance of the almost homomorphism from a strict homomorphism can be shown to depend not just on the degree of its nonhomomorphy but also on some "continuity scale.
Sobolev spaces and uniform boundary representations
No full textWe prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces with p>1 for arbitrary Gromov hyperbolic groups. These are closed subspaces of and in particular Hilbert spaces in the case .This construction allows us, for an appropriate choice of , to approximate the trivial representation through uniformly bounded representations. This phenomenon does not have analogue in the setting of isometricrepresentations whenever the hyperbolic group considered has the Property (T).The key is the introduction of a notion of metrically conformal operator on a metric space endowed with a conformal structure \`{a} la Mineyev and a metric analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform
A metric approach to limit operators
No full textWe extend the limit operator machinery of Rabinovich, Roch, and Silbermann from Z^N to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property
Coarse medians and Property A
No full textWe prove that uniformly locally finite quasigeodesic coarse median spaces of finite rank and at most exponential growth have Property A. This offers an alternative proof of the fact that mapping class groups have property A
Maximal and reduced Roe algebras of coarsely embeddable spaces
No full textIn [7], Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds into a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two results are natural coarse-geometric analogues of certain well-known implications of a-T-menability and amenability for group C*-algebras. The techniques used are E-theoretic, building on work of Higson, Kasparov and Trout [11], [12] and Yu [28]
Strong hyperbolicity
Full text linkWe propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(?1) spaces are strongly hyperbolic. On the way, we determine the best constant of hyperbolicity for the standard hyperbolic plane H2. We also show that the Green metric defined by a random walk on a hyperbolic group is strongly hyperbolic. A measure-theoretic consequence at the boundary is that the harmonic measure defined by a random walk is a visual Hausdorff measur
Measured expanders
No full textBy measured graphs, we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincaré inequalities. We prove that the so-called Cheeger inequality holds in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Moreover, we also prove that our measured (asymptotic) expanders are generalised expanders introduced by Tessera. Finally, we present some examples to demonstrate relations and differences between classical expander graphs and the measured ones. This paper is motivated primarily by our previous work on the rigidity problem for Roe algebras
Coarse non-amenability and coarse embeddings
No full textWe construct the first example of a coarsely non-amenable (= without Guoliang Yu’s property A) metric space with bounded geometry which coarsely embeds into a Hilbert space
Controlled coarse homology and isoperimetric inequalities
No full textWe study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize the existence of primitives of volume forms with prescribed growth and show that coarse homology classes obstruct weighted Poincaré inequalities
- …
