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

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arxiv 2209.08087 v2 pith:AR42W7UQ submitted 2022-09-16 math.GR math.ATmath.DSmath.KTmath.OA

classification math.GRmath.ATmath.DSmath.KTmath.OA
keywords groupsamplefullgroupoidstopologicalhomologyalgebraicgeneral
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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.

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  1. Homology of Steinberg algebras

    math.KT 2024-12 conditional novelty 7.0 of 10

    Hochschild and cyclic homology of Steinberg algebras of ample groupoids decompose in terms of groupoid homology, and for Exel-Pardo algebras these invariants and K-theory are computed by explicit cones and exact sequences.

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