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Long thin covers and nuclear dimension

Ilan Hirshberg, Jianchao Wu

Abstract

We establish finite nuclear dimension for crossed product C*-algebras arising from various classes of possibly non-free topological actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces, certain amenable actions of hyperbolic groups, and certain allosteric actions of wreath products. We obtain these results by introducing a new notion of dimension for topological dynamical systems, called the long thin covering dimension, which involves a suitable version of Rokhlin-type towers with controlled overlaps for possibly non-free actions.

Long thin covers and nuclear dimension

Abstract

We establish finite nuclear dimension for crossed product C*-algebras arising from various classes of possibly non-free topological actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces, certain amenable actions of hyperbolic groups, and certain allosteric actions of wreath products. We obtain these results by introducing a new notion of dimension for topological dynamical systems, called the long thin covering dimension, which involves a suitable version of Rokhlin-type towers with controlled overlaps for possibly non-free actions.
Paper Structure (25 sections, 66 theorems, 168 equations)

This paper contains 25 sections, 66 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Rokhlin-type towers for non-free actions
  4. Ideas behind the main theorem: "nearsighted stabilizers"
  5. Ideas behind the main theorem: LTC dimension recast
  6. Ideas behind the main theorem: the coarse orbit space
  7. Finite bounds on the long thin covering dimension
  8. Organization of the paper
  9. Preliminaries
  10. Covering dimension
  11. Asymptotic dimension for coarse spaces
  12. $C_0(X)$-algebras
  13. Nuclear dimension
  14. Virtually nilpotent and polycyclic groups
  15. The orbit coarse structure and its asymptotic dimension
  16. ...and 10 more sections

Key Result

Theorem 2.5

If $M$ is a subspace of a totally normal spaceA topological space is totally normal if any subspace is normal.$X$, then $\dim (M) \leq \dim (X)$.

Theorems & Definitions (173)

  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Pears75
  • Proposition 2.6: Pears75
  • Lemma 2.7: Engelking
  • Lemma 2.8: cf. Hirshberg-Wu16
  • Definition 2.9
  • Definition 2.10: HigsonPedersenRoe1997C, see also Roe2003Lectures
  • Example 2.11
  • Definition 2.12
  • ...and 163 more