Minimal surjective local homeomorphisms of finite-dimensional compact spaces give Deaconu-Renault groupoids with dynamical comparison, yielding UCT Kirchberg algebras and, in the Cantor case, Matui's AH conjecture.
Amenability and comparison for \'etale groupoids with polynomial growth
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abstract
We show that any second-countable locally compact Hausdorff \'etale groupoid with polynomial growth is topologically amenable. If moreover the groupoid is compactly generated with compact and metrizable unit space, it has weak $m$-comparison. Thus if the groupoid is also ample and minimal, it satisfies Matui's AH-conjecture.
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2026 1verdicts
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Dynamical comparison for local homeomorphisms
Minimal surjective local homeomorphisms of finite-dimensional compact spaces give Deaconu-Renault groupoids with dynamical comparison, yielding UCT Kirchberg algebras and, in the Cantor case, Matui's AH conjecture.