# Almost finiteness for general etale groupoids and its applications to stable rank of crossed products

## TL;DR

This paper generalizes the concept of almost finiteness to etale groupoids and demonstrates that their associated C*-algebras have stable rank one, with implications for crossed products of various group actions.

## Contribution

It introduces a new approach to almost finiteness for etale groupoids and proves stable rank one for their reduced C*-algebras, extending previous results.

## Key findings

- Reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one.
- Crossed products of certain group actions also have stable rank one.
- New local strategy for proving stable rank one in C*-algebras.

## Abstract

We extend Matui's notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whose minimal extensions form the crossed product of stable rank one. (iii) For any amenable group, the crossed product of the universal minimal action has stable rank one.