Continuity of critical exponent of quasiconvex-cocompact groups under Gromov-Hausdorff convergence

The paper proves that GCBqc(P0,r0,δ;D) is compact for pointed equivariant GH convergence and that the critical exponent is continuous (Theorem A) by combining closure under ultralimits, convergence of boundaries/limit sets, and a quantified Ahlfors regularity theorem that yields uniform control of boundary Minkowski dimension and orbit growth, culminating in Theorem 4.3 asserting h_{Γ_ω} = ω–lim h_{Γ_n} . The candidate solution’s overall plan mirrors the result but relies critically on an unjustified uniform bounded-multiplicity claim for almost-equivariant correspondences (its Step 3.2), which ignores the dependency on (P0,r0,δ,D) and is false without invoking a uniform free-systole (provided in the paper via Proposition 3.8) . It also cites an equivariant compactness theorem (Fukaya–Yamaguchi) outside the paper’s metric setting, whereas the paper uses closure under ultralimits specific to GCB spaces. Therefore, the paper’s argument is correct, while the model’s proof sketch is flawed.