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Strong relative Novikov conjecture for coarsely embeddable groups

Geng Tian, Zhizhang Xie, Guoliang Yu

Abstract

In this article, we prove a strong relative Novikov conjecture for any pair of groups that are coarsely embeddable into Hilbert space.

Strong relative Novikov conjecture for coarsely embeddable groups

Abstract

In this article, we prove a strong relative Novikov conjecture for any pair of groups that are coarsely embeddable into Hilbert space.
Paper Structure (11 sections, 20 theorems, 156 equations, 5 figures)

This paper contains 11 sections, 20 theorems, 156 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Roe algebras and localization algebras
  4. Equivariantly uniformly asymptotic morphisms
  5. $KK$-theory and its relation with $E$-theory
  6. Relative $\gamma$-reduced group $C^*$-algebras
  7. Relative $K$-homology and relative Baum-Connes assembly map
  8. Relative K-homology
  9. Relative Baum-Connes assembly map
  10. The relative Bott map
  11. Main results

Key Result

Theorem 1.2

Let $h\colon G\rightarrow \Gamma$ be a group homomorphism between two countable discrete groups. If both $G$ and $\Gamma$ are coarsely embeddable into Hilbert space, then the strong relative Novikov conjecture holds for $(G, \Gamma, h)$, that is, the relative Baum--Connes assembly map is injective.

Figures (5)

  • Figure 1: A commutative diagram
  • Figure 2: Asymptotic maps $h_\gamma$ from line \ref{['eq:homogamma']} and $h_\alpha^{\mathcal{A}}$ from line \ref{['eq:homoalphaA']}
  • Figure 3: Asymptotic maps $h_\gamma$ from line \ref{['eq:homogammaRoe']} and $h_\alpha^{\mathcal{A}}$ from line \ref{['eq:homoalphaARoe']}
  • Figure 4: The Asymptotic maps $h^L_\gamma$ from line \ref{['eq:localgamma']} and $h_\alpha^{L, \mathcal{A}}$ from line \ref{['eq:localhomoA']}
  • Figure 5: The asymptotic map $h^{L, \mathcal{A}}_\alpha$ from line \ref{['eq:localhomoA']} and the $C^*$ homomorphism $h_{\mathrm{max}, L}$ from line \ref{['eq:localhomo']}

Theorems & Definitions (59)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Gong-Wang-Yu
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 49 more