COST OF INNER AMENABLE GROUPOIDS

The paper’s main theorem explicitly proves that every principal extension of an inner amenable discrete p.m.p. groupoid has cost 1 (Theorem 1.1 and Theorem 4.4), using an ergodic reduction, a finite-classes case via isotropy groups and q-normality, and an aperiodic case built from a maximal amenable subgroupoid, a q-normal chain, and monotonicity of cost; see Theorem 1.1 and Theorem 4.4 together with Propositions 2.24, 2.27, and 2.29 and Theorem 4.3 . The candidate solution follows the same structure and invokes the same key ingredients (ergodic decomposition, maximal amenable subgroupoid, insertion of an intermediate q-normal subgroupoid, preimage-preservation of q-normality, and cost monotonicity), arriving at the same conclusion. Minor differences are present only in expository choices and a few citation mislabels, not in substance.