FIBERWISE AMENABILITY OF AMPLE ÉTALE GROUPOIDS

The paper proves that [[G]] is sofic when G is second countable, minimal, and admits a Følner sequence, via a compressed sofic representation built on finite subsets P_n ⊂ G(0) obtained from normal Følner sets and a key intersection lemma. The candidate solution proposes a different microstate construction acting on arrows and relies on a diagonal refinement and a simultaneous choice of u_l ∈ U_l avoiding a finite union of clopen sets. That selection step is not justified and can fail: a finite union of proper clopens can cover U_l, so the required u_l may not exist. Without this, the fixed-point control for nontrivial elements is not established. The paper’s argument is coherent and correctly applies Elek’s compressed sofic criterion; the model has a substantive gap in Step 2.