K-homological finiteness and hyperbolic groups

Emerson, Heath; Nica, Bogdan

doi:10.1515/crelle-2015-0115

Cited by 10 publications

(16 citation statements)

References 36 publications

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“…For torsion free Γ and M := X/Γ, there is a Morita equivalence C 0 (M ) ∼ C 0 (X) ⋊ r Γ, and a KK-equivalence C(H) ⋊ r Γ ∼ C * r (Γ). The exact sequence takes the form · · · → K * (C 0 (M )) → K * (C * r (Γ)) → K * (C(∂H) ⋊ r Γ) → · · · , as in [7,8]. 5.9.…”

Section: Corollarymentioning

confidence: 99%

Hecke modules for arithmetic groups via bivariant K-theory

Mesland

Sengun

2018

Ann. K-Th.

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Let Γ be a lattice in a locally compact group G. In another work, we used KK-theory to equip with Hecke operators the K-groups of any Γ-C * -algebra on which the commensurator of Γ acts. When Γ is arithmetic, this gives Hecke operators on the K-theory of certain C * -algebras that are naturally associated with Γ. In this paper, we first study the topological K-theory of the arithmetic manifold associated to Γ. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the KK-groups associated to an arithmetic group Γ become true Hecke modules. We conclude by discussing Hecke equivariant maps in KK-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with Γ. Along the way we discuss the relation between the Ktheory and the integral cohomology of low-dimensional manifolds as Hecke modules.

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Section: Corollarymentioning

confidence: 99%

Hecke modules for arithmetic groups via bivariant K-theory

Mesland

Sengun

2018

Ann. K-Th.

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“…and, using the canonical isomorphism C 0 (Γ) ⋉ Γ ∼ = K(l 2 Γ), and amenabity of the action, a result due to Adams in [1], we obtain a KK-class [∂ Γ ] ∈ KK Γ 1 (C(∂G), C) = K −1 (C(∂Γ) ⋊ r Γ). Alternatively, in [16] a Γ-equivariant completely positive splitting of (7.1) is provided, which implies the extension determines a KK 1 -class.…”

Section: Dirac Classes For Boundary Actions Of Negative Curved Groupsmentioning

confidence: 99%

The class of a fibre in noncommutative geometry

Emerson

2020

Journal of Geometry and Physics

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This note addresses the K-homology of a C*-algebra crossed product of a discrete group acting smoothly on a manifold, with the goal of better understanding its noncommutative geometry. The Baum-Connes apparatus is the main tool. Examples suggest that the correct notion of the 'Dirac class' of such a noncommutative space is the image under the equivalence determined by Baum-Connes of the fibre of the canonical fibration of the Borel space associated to the action, and a smooth model for the classifying space of the group. We give a systematic study of such fibre, or 'Dirac classes,' with applications to the construction of interesting spectral triples, and computation of their K-theory functionals, and we prove in particular that both the well-known deformation of the Dolbeault operator on the noncommutative torus, and the class of the boundary extension of a hyperbolic group, are both Dirac classes in this sense and therefore can be treated topologically in the same way.

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“…In particular, the obstructions to finite summability of spectral triples remain when generalizing to higher order as well as to semifinite spectral triples. The lack of finitely summable spectral triples does not preclude the existence of finitely summable Fredholm modules (see [25,26]), and we exploit this later. [19,20,34,12,13].…”

Section: Smoothness and Summabilitymentioning

confidence: 99%

Untwisting twisted spectral triples

2019

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We examine the index data associated to twisted spectral triples and higher order spectral triples. In particular, we show that a Lipschitz regular twisted spectral triple can always be "logarithmically dampened" through functional calculus, to obtain an ordinary (i.e. untwisted) spectral triple. The same procedure turns higher order spectral triples into spectral triples. We provide examples of highly regular twisted spectral triples with non-trivial index data for which Moscovici's ansatz for a twisted local index formula is identically zero. *

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K-homological finiteness and hyperbolic groups

Abstract: Abstract Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a \mathrm{C}^{*} … Show more

Cited by 10 publications

References 36 publications

Hecke modules for arithmetic groups via bivariant K-theory

Hecke modules for arithmetic groups via bivariant K-theory

The class of a fibre in noncommutative geometry

Untwisting twisted spectral triples

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