Sigma-compact etale groupoids receive a canonical coarse metric; the new fiberwise amenability notions recover group and metric amenability, imply invariant measures, coincide for minimal groupoids, and yield a Følner-paradoxical dichotomy.
Amenability of coarse spaces and
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Fiberwise amenability of \'{e}tale groupoids
Sigma-compact etale groupoids receive a canonical coarse metric; the new fiberwise amenability notions recover group and metric amenability, imply invariant measures, coincide for minimal groupoids, and yield a Følner-paradoxical dichotomy.