Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
Xin Li
TL;DR
The paper builds a comprehensive framework unifying ample groupoids, topological full groups, and algebraic K-theory spectra through small permutative categories of bisections, linking groupoid homology to the homology of topological full groups via infinite loop spaces. Under mild hypotheses (minimality, unit space without isolated points, and comparison), it proves that $H_*(\mathbf{F}(G)) \cong H_*(\Omega^{\infty}_0 \mathbb{K}(\mathfrak{B}_G))$ and establishes the foundational isomorphism $\tilde{H}_*(\mathbb{K}(\mathfrak{B}_G)) \cong H_*(G)$, enabling complete rational computations and broad vanishing results. The work also extends Matui's AH-conjecture to broad classes of ample groupoids via Morita invariance and the Atiyah–Hirzebruch spectral sequence, and develops a robust machinery for comparing amplified and original topological full groups. These results yield new insights into Thompson-type groups, SFT and tiling groupoids, and self-similar action groupoids, including the construction of continuum many pairwise non-isomorphic integrally acyclic infinite simple groups. Overall, the framework provides a powerful, topologically flavored toolkit for understanding homology in this dynamic-algebraic setting with wide-ranging applications.
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.