Group quasi-representations  and almost flat bundles

Dǎdǎrlat, Marius

doi:10.4171/jncg/152

Cited by 17 publications

(26 citation statements)

References 28 publications

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“…What sort of groups have this weaker property (which should also imply the strong Novikov conjecture)? Recent work of Dadarlat [14,15] 10 investigating assembly and quasi-representations (among other things) seems very relevant here.…”

Section: Questions 321mentioning

confidence: 99%

A finite-dimensional approach to the strong Novikov conjecture

Ramras

Willett

2013

Algebr. Geom. Topol.

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A finite dimensional approach to the strong Novikov conjecture DANIEL RAMRAS RUFUS WILLETT GUOLIANG YUThe aim of this paper is to describe an approach to the (strong) Novikov conjecture based on continuous families of finite dimensional representations: this is partly inspired by ideas of Lusztig related to the Atiyah-Singer families index theorem, and partly by Carlsson's deformation K -theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the K -theory and cohomology of representation spaces.19K56, 55N15, 57R20; 20C99, 46L85, 46L80 IntroductionThe aim of this paper is to study the strong Novikov conjecture [19] for a finitely presented group Γ. If we assume that Γ has a finite classifying space BΓ, one version of this conjecture states that the analytic assembly mapis rationally injective; here the left hand side is the K -homology of BΓ and the right hand side is the K -theory of the maximal group C * -algebra of Γ. We give a definition of the analytic assembly map in Section 2 below. The strong Novikov conjecture implies the classical Novikov conjecture on homotopy invariance of higher signatures, as well as being closely related to several other famous conjectures. A naive version of our approach might proceed as follows. A finite dimensional unitary representationof Γ defines a vector bundle E ρ over BΓ via a well-known balanced product construction. E ρ defines an element [E ρ ] of the K -theory group K * (BΓ) and thus a 'detecting homomorphism'As is well-known 1 , ρ * factors through the analytic assembly map µ; hence µ(x) = 0 for any x ∈ K * (BΓ) such that there exists ρ : Γ → U(n) with ρ * (x) = 0. Thus if one can find 'enough' representations to detect all of K * (BΓ), one would have proved the strong Novikov conjecture.Unfortunately, this approach will not work: the bundles E ρ are flat, so Chern-Weil theory tells us that any 'detecting homomorphism' as in line (2) above is rationally trivial on reduced K -homology. One possible way to salvage the idea in the paragraph above is to use infinite dimensional representations. This led to the Fredholm representations of Miscenko [23], and subsequently to Kasparov's KK-theory [19]; both of these, and the closely related approach to the Novikov conjecture through the Baum-Connes conjecture [7] have proved enormously fruitful.In this paper we suggest a different approach. The central idea is not to use a single representation as in line (1) above, but instead a continuous family of representationsparametrized by a topological space X . Such a family defines a bundle E ρ over X × BΓ and thus a detecting homomorphismfrom the K -homology of BΓ to the K -theory of X via slant product with [E ρ ] ∈ K * (BΓ × X). This ρ * still factors through the analytic assembly map. The central result of this paper is that for...

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Section: Questions 321mentioning

confidence: 99%

A finite-dimensional approach to the strong Novikov conjecture

Ramras

Willett

2013

Algebr. Geom. Topol.

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“…3.2). Moreover, we show that this group provides a counterexample to a conjecture from [8]. Specifically, we prove that the natural map [[I(G), K]] → K 0 (I(G)) is not an isomorphism (Lem.…”

Section: Introductionmentioning

confidence: 89%

“…It was conjectured in [8] that if G is a torsion free discrete amenable group, then [[I(G), K]] ∼ = KK(I(G), C). We argue now that this conjecture fails for the Hantzsche-Wendt group.…”

Section: -Orbitsmentioning

confidence: 99%

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Connective C⁎-algebras

Dǎdǎrlat

Pennig

2017

Journal of Functional Analysis

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Connectivity is a homotopy invariant property of separable C * -algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C * -algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C *algebras. We give new characterizations of connectivity for exact and for nuclear separable C * -algebras and show that an extension of connective separable nuclear C * -algebras is connective. We establish connectivity or lack of connectivity for C * -algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups.

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“…It is implicitly conjectured in [8] that the augmentation ideal I(G) of a discrete, torsion free, amenable group G is homotopy symmetric and hence that the Kasparov group KK(I(G), B) can be realized as the homotopy classes of asymptotic morphisms [[I(G), B ⊗ K]] for any separable C * -algebra B. The case of abelian groups is covered by results from [9].…”

Section: Introductionmentioning

confidence: 99%

Deformations of wreath products

Dǎdǎrlat

Pennig

Schneider

2016

Bull. London Math. Soc.

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Connectivity is a homotopy invariant property of a separable C * -algebra A which has three important consequences: absence of nontrivial projections, quasidiagonality and realization of the Kasparov group KK(A, B) as homotopy classes of asymptotic morphisms from A to B ⊗ K if A is nuclear. Here we give a new characterization of connectivity for separable exact C*-algebras and use this characterization to show that the class of discrete countable amenable groups whose augmentation ideals are connective is closed under generalized wreath products. In a related circle of ideas, we give a result on quasidiagonality of reduced crossed-product C*-algebras associated to noncommutative Bernoulli actions.

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Group quasi-representations and almost flat bundles

Cited by 17 publications

References 28 publications

A finite-dimensional approach to the strong Novikov conjecture

A finite-dimensional approach to the strong Novikov conjecture

Connective C⁎-algebras

Deformations of wreath products

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