Groupoid homology and K-theory for algebraic actions from number theory

TL;DR

This paper computes the homology and K-theory of groupoids and C*-algebras associated with algebraic actions from rings of algebraic integers and integral dynamics, providing new calculations and resolving existing conjectures.

Contribution

It introduces new methods to compute groupoid homology and K-theory for algebraic actions, avoiding duality techniques and resolving a conjecture in the field.

Findings

Computed groupoid homology for algebraic action groupoids

Derived K-theory for ring C*-algebras of algebraic integers

Resolved the conjecture on K-theory for integral dynamics C*-algebras

Abstract

We compute the groupoid homology for the ample groupoids associated with algebraic actions from rings of algebraic integers and integral dynamics. We derive results for the homology of the topological full groups associated with rings of algebraic integers, and we use our groupoid homology calculation to compute the K-theory for ring C*-algebras of rings of algebraic integers, recovering the results of Cuntz and Li and of Li and L\"uck without using Cuntz--Li duality. Moreover, we compute the K-theory for C*-algebras attached to integral dynamics, resolving the conjecture by Barlak, Omland, and Stammeier in full generality.