Patterson-Sullivan theory for groups with a strongly contracting element

The paper proves the two claimed inequalities using a Patterson–Sullivan framework adapted to actions with a strongly contracting element, together with a clean Hilbert-space (l2 over G/N) argument for the G/N term and an ergodicity/uniqueness argument on the reduced horoboundary to force the strict half-gap under divergence. By contrast, the model’s Part I hinges on an unproven “two-sided normal form” g ≈ n1·σ([g])·n2 with uniformly bounded multiplicity and a derived convolution lower bound P_G(s) ≥ const·P_N(s)^2·P_{G/N}(2s). This factorization and the resulting inequality are not established in the paper and are generally unjustified in the stated generality. The model’s Part II gestures at the right strategy for strictness but omits key measure-theoretic ingredients the paper uses (reduced horoboundary, contracting limit set, and almost-uniqueness), so the model proof is incomplete.