FULL REALIZATION OF ERGODIC IRS ENTROPY IN SL2(Z) AND FREE GROUPS

The paper proves full IRS spectrum for free groups and virtually free groups under an adapted, symmetric random walk with finite 4th moment (Theorems 1.4–1.5) and develops the extension from a finite-index free subgroup via the hitting measure and core inequalities, not by a direct 1/[G:F] scaling of IRS entropies after lifting. The candidate solution misstates key hypotheses and steps: (i) it claims Ron‑George–Yadin proved the free-group result under finite second moment, whereas the paper requires finite 4th moment and explicitly remarks that extending to second moment is conjectural (Theorem 1.4 and Remark 1.6) ; (ii) it reverses the p→0 and p→1 endpoints for intersectional IRSs (Proposition 2.1 shows λp,K→δCore∅ as p→0, yielding zero entropy, and λp,K→δCoreG as p→1) ; and (iii) it asserts an unproven exact 1/[G:F] scaling of IRS entropies under lifting from F to G. The paper instead uses Proposition 2.5 (Abramov-type equality for quotient entropies), Lemma 2.7 (a ≥ [G:F] inequality at the level of cores), and continuity/approximation to realize the whole interval on G . The Poisson-bundle identification of IRS entropy with Furstenberg entropy is correctly invoked by both sides in spirit, but the candidate’s derivation for the virtually free case relies on a scaling equality that is not established in the paper. Overall, the paper’s argument is correct and complete, while the model’s proof contains substantive inaccuracies.