Failure of rational injectivity for minimal principal étale groupoids

Applying OpenAI's torsion-free group counterexample, we construct a second-countable locally compact Hausdorff minimal principal ample groupoid with Cantor unit space whose ordinary reduced Baum--Connes assembly map has an infinite-order kernel class in degree zero. Its rationalization is therefore not injective, with the natural coefficient algebra of continuous functions on the unit space. The groupoid is a transformation groupoid of a finitely generated torsion-free group and is compactly generated. We use the invariant-measure trace class of Antonini, Azzali and Skandalis to preserve an ordinary assembly kernel class under extension of coefficients to continuous functions. Elek's free minimal measure-preserving Cantor action supplies the required dynamics. The standard compact-action comparison identifies group assembly with coefficients and transformation-groupoid assembly. Thus minimality and principality do not restore the natural-coefficient reduced Baum--Connes assertion.

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