We describe a correspondence between spaces with walls and CAT(0) cube complexes.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-16.abs.html Version 3: minor misprint correcte

Abstract. We show that every non-elementary hyperbolic group Γ admits a proper affine isometric action on L p (∂Γ × ∂Γ), where ∂Γ denotes the boundary of Γ and p is large enough. Our construction involves a Γ-invariant measure on ∂Γ×∂Γ analogous to the Bowen-Margulis measure from the CAT(−1) setting, as well as a geometric, Busemann-type cocycle. We also deduce that Γ admits a proper affine isometric action on the first ℓ p -cohomology group H 1 (p) (Γ) for large enough p.

Spectral morphisms between Banach algebras are useful for comparing their K-theory and their "noncommutative dimensions" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce.

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Abstract Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a \mathrm{C}^{*} -algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of \mathrm{C}^{*} -algebras associated to hyperbolic groups – the \mathrm{C}^{*} -crossed product for the boundary action, and the reduced group \mathrm{C}^{*} -algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.