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Nuclear dimension and virtually polycyclic groups

Caleb Eckhardt, Jianchao Wu

TL;DR

The paper studies when group C*-algebras have finite nuclear dimension, linking this property to Hirsch length via virtually polycyclic and elementary amenable groups. The authors develop a framework based on central extensions and twisted crossed products, applying the Hirshberg–Wu analysis of actions of virtually nilpotent groups on $C_0(X)$-algebras, and an inductive scheme on Hirsch length to obtain uniform finite bounds. They prove finite nuclear dimension for twisted group C*-algebras $C^*(G,\sigma)$ when $G$ is virtually polycyclic, and extend the finite-nuclear-dimension phenomenon to a broader class of elementary amenable groups, including many non-residually finite wreath products, while formulating a parallel conjecture for decomposition rank. Conversely, they exhibit large wreath products with infinite Hirsch length that yield infinite nuclear dimension, thereby delineating the boundary between finite and infinite dimension in terms of group structure. The results illuminate a deep connection between group-theoretic dimension notions and noncommutative dimension theories, providing new finite-nuclei examples beyond residually finite groups and suggesting a broader classification program for C*-algebras arising from groups.

Abstract

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on $C_0(X)$-algebras.

Nuclear dimension and virtually polycyclic groups

TL;DR

The paper studies when group C*-algebras have finite nuclear dimension, linking this property to Hirsch length via virtually polycyclic and elementary amenable groups. The authors develop a framework based on central extensions and twisted crossed products, applying the Hirshberg–Wu analysis of actions of virtually nilpotent groups on -algebras, and an inductive scheme on Hirsch length to obtain uniform finite bounds. They prove finite nuclear dimension for twisted group C*-algebras when is virtually polycyclic, and extend the finite-nuclear-dimension phenomenon to a broader class of elementary amenable groups, including many non-residually finite wreath products, while formulating a parallel conjecture for decomposition rank. Conversely, they exhibit large wreath products with infinite Hirsch length that yield infinite nuclear dimension, thereby delineating the boundary between finite and infinite dimension in terms of group structure. The results illuminate a deep connection between group-theoretic dimension notions and noncommutative dimension theories, providing new finite-nuclei examples beyond residually finite groups and suggesting a broader classification program for C*-algebras arising from groups.

Abstract

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on -algebras.
Paper Structure (21 sections, 36 theorems, 103 equations)

This paper contains 21 sections, 36 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Outline of Proof
  3. Organization of the Paper
  4. Acknowledgments
  5. Preliminaries
  6. Nuclear dimension
  7. Polycyclic groups
  8. Group cohomology
  9. Twisted crossed products
  10. The stabilization trick
  11. Twisted $G$-$X$-C*-algebras
  12. Twisted group C*-algebras
  13. Twisted group C*-algebras as twisted crossed products
  14. Twisted group C*-algebras and central extensions
  15. A useful class of C*-algebras
  16. ...and 6 more sections

Key Result

Theorem A

Let $G$ be a virtually polycyclic group and $\sigma$ a circle-valued 2-cocycle on $G$. Then the twisted group C*-algebra $C^*(G,\sigma)$ has finite nuclear dimension. In the case of untwisted group C*-algebras of such groups, there is a finite bound on their nuclear dimension that depends only on th

Theorems & Definitions (84)

  • Theorem A: see Theorems \ref{['thm:finitend']} and \ref{['thm:twistednucdim']}
  • Conjecture I
  • Theorem B: see \ref{['thm:nonRFexample']}
  • Theorem C
  • Conjecture II
  • Theorem D
  • proof
  • Definition 2.1
  • Lemma 2.2: see WinterZacharias2010nuclear and Carrion11
  • Lemma 2.3
  • ...and 74 more