Nuclear dimension and virtually polycyclic groups

TL;DR

This paper proves that group C*-algebras of virtually polycyclic groups have finite nuclear dimension, introduces related conjectures, and verifies them for certain elementary amenable groups, including new examples of non-residually finite groups.

Contribution

It establishes finite nuclear dimension for virtually polycyclic group C*-algebras and verifies conjectures for broader classes of elementary amenable groups, including non-residually finite examples.

Findings

Finite nuclear dimension for virtually polycyclic group C*-algebras.

Verification of conjectures for elementary amenable groups.

First examples of finitely generated, non-residually finite groups with finite nuclear dimension.

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.