Exactness and amenability at infinity for locally compact groupoids

For a second-countable locally compact Hausdorff étale groupoid G, we prove the equivalence of exactness of Cr*(G), exactness of reduced crossed products, and strong amenability at infinity. These conditions are also equivalent to exactness of any one twisted reduced algebra. The amenable action can be chosen with a proper open surjective anchor admitting a continuous section. This answers Anantharaman–Delaroche’s 2016 exactness questions and the étale case of LaLonde’s 2017 question about exact transformation groupoids. Consequently, exactness together with canonical weak containment implies amenability; for compact unit space, exactness is equivalent to amenability of the Furstenberg boundary action. We also prove permanence of strong amenability at infinity under continuous strongly surjective homomorphisms between second-countable groupoids with Haar systems, when the kernel and target have this property. No Haar system on the kernel is required. Finally, we construct an inner-exact étale group bundle with exact fibers and nonexact reduced algebra, supplying the example asked for by Anantharaman–Delaroche in 2016 and answering LaLonde’s 2017 group-bundle question negatively. Its fibers are hyperbolic groups converging strongly to a free group, while their Cayley graphs fail uniform property A.

GitHub repository: https://github.com/eulogica/groupoid-exactness-amenability