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Metric sparsification and operator norm localization. (English) Zbl 1148.46040
For a finitely generated residually finite group \(\Gamma\) with a sequence of normal subgroups \(\Gamma_1 \supseteq \Gamma_2 \supseteq \dots\) such that \(\bigcap_{i=1}^\infty \Gamma_1 = \{e\}\) and with the quotient metric on \(\Gamma/\Gamma_i\), defined by \(d(a\Gamma_i,b\Gamma_i) := \min\{ d(a\gamma_1,b\gamma_2) ; \gamma_1,\gamma_2\in \Gamma_i\}\), one defines the so called box metric space \(X(\Gamma):= \amalg_{i=1}^\infty \Gamma/\Gamma_i\) as the disjoint union of \(\Gamma/\Gamma_i\) with the metric such that the restrictions to \(\Gamma/\Gamma_i\) coincide with their quotient metric and that \(\lim_{n+m\to\infty, n\neq m} d(\Gamma/\Gamma_n,\Gamma/\Gamma_m) = \infty\), see e.g. [J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser. 31, Am. Math. Soc. (2003; Zbl 1042.53027)]. Denote by \(E\Gamma\) the universal space for the free and proper \(\Gamma\) actions and \(C^*_r(\Gamma)\) the reduced group C*-algebra. The Strong Novikov Conjecture states that the Baum-Connes map \(\mu_r : K^\Gamma_*(E\Gamma) \to K_*(C^*_r(\Gamma))\) is injective. If \(X\) is a discrete metric space with bounded geometry, the Coarse Geometric Novikov Conjecture states that the Baum-Connes map \(\mu : \lim_{d\to\infty} K_*(P_d(X)) \to K_*(C^*(X))\) is injective, see e.g. [G. Yu, Invent. Math. 139, No. 1, 201–240 (2000; Zbl 0956.19004)], where \(P_d(X)\) is the Rips complex and \(C^*(X)\) is the Roe algebra associated to \(X\).
In the paper under review, the authors study an operator norm localization property (Definition 2.2) and its applications to the coarse Novikov conjecture in operator K-theory. The authors introduce some sparsification property (Definition 3.1) which is (Proposition 4.1) a sufficient geometric condition for the operator norm localization property. This leads the authors to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. In §6 they show that a sequence of expanding graphs does not posseses the operator norm localization property.
If \(\Gamma\) has the operator norm localization property and the classifying space \(E\Gamma/\Gamma\) for free \(\Gamma\)-actions has homotopy type of a compact CW complex, then the strong Novikov conjecture for \(\Gamma\) and \(\Gamma_n (n=1,2,3,...)\) implies the coarse geometric Novikov conjecture for \(X(\Gamma)\) (Theorem 7.1).

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K35 Kasparov theory (\(KK\)-theory)
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