Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

TL;DR

This paper establishes a broad framework connecting ample groupoids, topological full groups, algebraic K-theory spectra, and infinite loop spaces, enabling new insights into their homological properties and applications.

Contribution

It introduces a novel construction of permutative categories of compact open bisections to analyze homological invariants of topological full groups.

Findings

Complete rational computations of homology

Vanishing and acyclicity results for group homology

Proof of Matui's AH-conjecture for minimal, ample groupoids

Abstract

Inspired by work of Szymik and Wahl on the homology of Higman-Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids. Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui's AH-conjecture for all minimal, ample groupoids with comparison.