ON THE LATTICE OF BOUNDARIES AND THE ENTROPY SPECTRUM OF HYPERBOLIC GROUPS

The paper states Theorem B with the strong join identity B_α ∨ B_β = B_{α∧β}, but in the proof of §6 it only constructs a tree of hyperbolic quotients and shows distinctness/embedding; it does not justify the join identity or the kernel-equality it would require. The proof ends by observing distinct essential stabilizers, concluding only that N^{<∞} embeds into BL(Γ, μ) (see Proposition 6.1 and the subsequent proof sketch of Theorem B) . The candidate solution repairs the property-(T) citation but relies on an unproven kernel–join lemma and misattributes a crucial kernel-intersection property to Proposition 6.1 (which does not assert it), and also incorrectly treats products of boundaries as boundaries. Hence both the paper and the model leave key steps unjustified.