Free actions of polynomial growth Lie groups and classifiable C*-algebras

TL;DR

This paper proves that free actions of polynomial growth Lie groups on finite spaces have finite nuclear dimension, leading to the classification of certain associated C*-algebras, including those from cut-and-project sets.

Contribution

It establishes finite nuclear dimension for crossed product C*-algebras arising from free polynomial growth Lie group actions, enabling their classification.

Findings

Finite tube dimension for free Lie group actions

Finite nuclear dimension of associated C*-algebras

Classifiability of C*-algebras from cut-and-project sets

Abstract

We show that any free action of a connected Lie group of polynomial growth on a finite dimensional locally compact space has finite tube dimension. This is shown to imply that the associated crossed product C*-algebra has finite nuclear dimension. As an application we show that C*-algebras associated with certain aperiodic point sets in connected Lie groups of polynomial growth are classifiable. Examples include cut-and-project sets constructed from irreducible lattices in products of connected nilpotent Lie groups.