{"id":"69f8a37f-ff6f-4067-bab4-180d59c10e9e","arxiv_id":"2507.17466","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Random finite covers of manifolds with Ricci curvature bounded below have no new Laplacian eigenvalues in [0,Λ] when the fundamental group satisfies strong convergence of permutation representations.","lead":"This paper proves that random finite coverings of certain curved spaces do not create new low-frequency vibrations below a geometric threshold. It generalizes recent spectral gap results from hyperbolic surfaces to manifolds with Ricci curvature bounded below.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 7.6 uses Section 6, whose standing assumption is bounded sectional curvature, while the main theorem assumes only Ricci bounded below; this hypothesis mismatch is the load-bearing gap.","rationale":"I read the paper as an attempt to generalize Hide-Magee spectral stability from hyperbolic surfaces to arbitrary complete Riemannian manifolds with Ricci curvature bounded below, provided the fundamental group satisfies (PRP)/(PRS). The proof strategy is sound in outline: construct an end parametrix using the essential-spectrum gap, an interior parametrix using the Cheng-Yau gradient estimate on the universal cover, then use strong convergence of permutation representations to control the error. The reader's weakest assumption, Proposition 4.5, concerns the uniform gradient bound for the resolvent kernel; I examined the verification and it appears correct: G_λ is a positive eigenfunction, the differential-inequality hypothesis of Theorem 2.13 holds with C = λ0(M̂), and the distance cutoff d(x,y) ≥ 1/2 avoids the singularity. The same holds for the λ-difference quotients in Proposition 5.2, up to a harmless typo in the displayed estimate. Thus I do not see a real analytic flaw there. The more concrete and verifiable problem is the hypothesis mismatch between Section 6 and Theorem 7.6. The manuscript explicitly states bounded sectional curvature at the start of Section 6 and never relaxes it, while the main theorem assumes only Ricci bounded below. Since (6.7) from Section 6 is the key parametrix identity used in the proof of Theorem 7.6, this is a load-bearing gap as written. I agree with the reader's conditional verdict: the result is likely correct and the gap likely removable, but the proof does not currently match the theorem statements. The concrete test I propose would settle whether the gap is purely cosmetic, in which case the verdict could later be upgraded to ACCEPT, or whether a genuine obstruction exists, in which case the main theorem is unproved in its claimed generality.","tokens_in":21511,"tokens_out":24623,"duration_ms":270139,"concrete_test":"Verify directly whether [15, Lemma 5.4] holds for an arbitrary complete Riemannian manifold with Ric ≥ -(m-1)b², focusing on the assertion in Proposition 6.4: for a compact D ⊂ F there exist compact D1 ⊂ F and finite S ⊂ Γ such that the integral kernel is supported in ∪_{γ∈S} γ^{-1}D and is zero unless x ∈ D1. Prove this from properness of complete manifolds and local finiteness of the deck action, without using sectional-curvature bounds. If the proof goes through, Section 6's standing assumption can be weakened to Ricci lower bound and the gap is cosmetic; if it genuinely needs bounded sectional curvature, then Theorem B is unproved for general Ricci-lower-bound manifolds and the verdict should be CONDITIONAL or REJECT accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires [0,Λ]-stability for manifolds with Ricci curvature bounded below only. However, Section 6 opens: \"Throughout this section let M be a complete Riemannian manifold with bounded sectional curvature.\" The main theorem (Theorem 7.6) then states the result under Ricci bounded below and its proof invokes (6.7) from Section 6 without ever discharging the stronger sectional-curvature hypothesis. The only step where the stronger assumption could be genuinely needed is the support/finite-propagation argument in Proposition 6.4, which cites [15, Lemma 5.4] and asserts that \"the proof of [15, Lemma 5.4] works in our setting without any changes.\" That lemma was designed for compact hyperbolic surfaces and may rely on properties of Dirichlet fundamental domains that are not automatic under Ricci lower bounds alone. If the lemma requires bounded sectional curvature, or cocompactness, the proof of Theorem 7.6 does not cover the stated generality. The gap is likely removable, since the needed finiteness should follow from properness of complete manifolds and local finiteness of the deck action, but the manuscript never provides that verification. As written, the proof of the central theorem rests on an unproven stronger assumption, which is a concrete correctness risk independent of the plausibility of the analytic estimates in Sections 4 and 5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general parametrix method, following Hide and Magee, to prove that finite coverings of a complete connected Riemannian manifold M are spectrally stable on [0, Λ] provided Λ is below min{λ_ess(M), λ_0(\\widetilde M)} and the fundamental group satisfies a strong-convergence condition (PRP) for permutation representations. The main technical theorem, Theorem 7.6, reduces [0, Λ]-stability to finitely many inequalities (1.2) for finitely supported matrix-valued maps; Theorems A, B, and D and the corollaries then follow from known (PRP)/(PRS) results for free groups, surface groups, and limit groups. The analytic core is a parametrix built from the resolvent on the universal cover using Cheng–Yau gradient estimates and a Cheeger–Colding cutoff, patched together with an end parametrix and an interior parametrix.","tokens_in":21650,"tokens_out":18334,"duration_ms":188204,"significance":"If the Section 6 hypothesis gap is resolved, the result is a substantial generalization of the uniform spectral gap results for hyperbolic surfaces to arbitrary complete manifolds with a Ricci lower bound and positive µ, contingent on the group-theoretic (PRP)/(PRS) inputs. The paper's main novelty is the elimination of explicit heat-kernel asymptotics on the universal cover by using Cheng–Yau estimates; this is a genuinely useful technical contribution. The dependence on external deep results—Bordenave–Collins [6], Magee–Puder–van Handel [22], and Louder–Magee [17]—is clearly stated, and the final theorems are sharply formulated. However, as written the proof of the central theorem relies on a standing assumption in Section 6 that is not assumed in the statements, so the verification is conditional.","major_comments":[{"comment":"Section 6 opens with the standing assumption that M has bounded sectional curvature, but Theorem 7.6 (and consequently Theorems A, B, D and the corollaries) is stated under only a Ricci lower bound. The proof of Theorem 7.6 uses (6.7), the operators L^int_φ(λ) and L^K_φ(λ), and the norm bounds obtained from Corollary 7.4, all of which are built inside Section 6; no paragraph discharges the stronger sectional-curvature hypothesis. In particular, Proposition 6.4 invokes [15, Lemma 5.4] via the assertion that its proof 'works in our setting without any changes'; that lemma is not stated, and the needed finite-support/finite-translate property for the kernel of R_{T,n}(λ)(1−χ^-_K) is load-bearing for the boundedness of the interior parametrix and for the reduction of L^T_φ(λ) to a finite sum over S in Section 7. As written, the main theorem is not proved at the stated level of generality. The authors should either prove the finiteness statement directly under the hypotheses of Theorem 7.6, using properness of complete M and local finiteness of the deck action, or explicitly restrict the main theorems to bounded sectional curvature.","section":"§6 and Theorem 7.6"},{"comment":"The displayed chain in the proof of Proposition 5.2 contains an unjustified step: after applying Theorem 2.13 to (∆−λ1)F, one obtains c1(∆−λ1)F + λ1∥∇F∥, and the line '≤ c1∆F + λ0∥∇F∥ − c1F' is not a valid consequence (the negative term should be −c1λ1F, and for λ1<1 there is no reason for the right-hand side to dominate the left-hand side). The intended bound can be recovered directly: since Gλ2=(∆−λ1)F ≤ ∆F and ∆F>0, one has Gλ2 ≤ ∆F ≤ m∥Hess F∥, so the hypotheses of Theorem 2.13 hold for F with the constant max(mc1, λ0). Please correct the proof, because Proposition 5.2 feeds Corollary 5.3, Lemma 7.1, and the uniform-in-λ estimate (7.5) used in Corollary 7.4.","section":"§5, proof of Proposition 5.2"}],"minor_comments":[{"comment":"In the estimate for ∥R_{T,n}(λ)(1−χ^-_K)f∥^2, the factor ∥R_T(λ)∥_{L^2} should be squared; the displayed inequality is dimensionally wrong and can fail when ∥R_T(λ)∥>1.","section":"§6, proof of Proposition 6.4"},{"comment":"Equation (2.1) is typeset with malformed norm symbols; it should read ∥∇f∥^2_{L^2} / ∥f∥^2_{L^2}.","section":"§2, Eq. (2.1)"},{"comment":"The reference to Cheeger–Colding appears as 'Theorem 6.3' in Section 1.2 and as 'Theorem 6.33' in Theorem 2.12; please make the numbering consistent.","section":"§1.2 and §2.3"},{"comment":"The application of Theorem 2.13 to G_λ is compressed; please state explicitly the ball radius (e.g., ρ=1/4 with balls centered at x) and the constant C=λ0(\\widetilde M) used in the hypotheses, so that the claimed independence of λ and T is transparent.","section":"§4, Proposition 4.5"}],"recommendation":"major_revision","confidential_remarks":"The geometric core is promising and the external-input structure is sound, but the hypothesis gap in Section 6 should be closed before publication. I would be happy to see a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a genuinely broader theorem: for any complete manifold with Ricci curvature bounded below and μ>0, random finite covers are [0,Λ]-stable if the fundamental group satisfies (PRP) or (PRS). This covers the hyperbolic-surface cases as special cases, but also higher-dimensional hyperbolic manifolds, Schottky manifolds, and quasi-Fuchsian threefolds. The key technical move—replacing explicit heat-kernel asymptotics with Cheng–Yau gradient estimates—is right and is what makes the generalization work. The parametrix construction follows Hide–Magee, and the representation-theoretic input (PRP)/(PRS) is external and well-established, which the paper states honestly.\n\nThe biggest soft spot is a real mismatch: Section 6 opens with the standing assumption \"bounded sectional curvature,\" while Theorem 7.6 states the result under Ricci bounded below only. The proof of Theorem 7.6 invokes (6.7) and Corollary 7.4, which come from Section 6, without discharging the stronger assumption. On reading, the actual use of bounded sectional curvature appears limited to the support argument in Proposition 6.4 and the analogous step in Section 7, both citing [15, Lemma 5.4]. That lemma should go through under properness of complete manifolds and local finiteness of the deck action; the needed finiteness is a consequence of the cutoff support and properness, not sectional curvature. So the gap is likely removable, but as written the proof does not literally cover the stated theorem. That needs to be fixed, at minimum by a remark that the support argument only needs properness.\n\nA second soft spot is Proposition 4.5: the uniform gradient estimate for the resolvent kernel is asserted to follow from Cheng–Yau, but the verification of the required differential inequality for G_λ and its λ-difference quotients is compressed. If the constant degenerates as λ approaches λ0(M), the parametrix bounds would collapse. I think it is probably fine, but it deserves scrutiny.\n\nThis paper is for spectral geometers and people working on random covers. It deserves a serious referee; with the sectional-curvature mismatch addressed, the result is likely correct and advances the field. I would send it to review.","headline":"A substantial generalization of random-cover spectral stability to all Ricci-lower-bounded manifolds, with a removable-looking but real hypothesis mismatch in the written proof.","tokens_in":22326,"tokens_out":2908,"would_cite":true,"duration_ms":30793,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J50","53C20","35P99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random finite coverings of a Ricci-bounded manifold almost surely add no new eigenvalues below the spectral thresholds, assuming the strong-convergence property (PRP) for the fundamental group.","keywords":["Laplace operator","spectral stability","spectral gap","random coverings","finite Riemannian coverings","Cheng-Yau gradient estimate","permutation representations","essential spectrum"],"falsifier":"On real hyperbolic space $\\mathbb{H}^m$ the Green's function is explicit, so Proposition 4.5 can be checked numerically: compute the supremum of $\\|\\nabla_x G_\\lambda(x,y)\\|/G_\\lambda(x,y)$ over $d(x,y)\\ge 1/2$ and $\\lambda\\in[0,1/4-\\delta]$, and verify that it is a finite constant depending only on $m$, the Ricci bound, and $1/4$; a divergence as $\\lambda\\uparrow 1/4$ would falsify the estimate and break the proof. To test the theorem itself, one would compute spectra of random covers of a fixed closed hyperbolic 3-manifold (whose fundamental group is free, hence (PRP) holds) for a fixed $\\Lambda<1/4$: if a new eigenvalue in $[0,\\Lambda]$ appears with probability bounded away from zero as the covering degree grows, Theorem B is false.","tokens_in":21181,"feed_emoji":"🎲","tokens_out":26610,"duration_ms":250951,"temperature":0.7,"pith_summary":"Random finite coverings of a manifold do not create new low-energy eigenvalues. Concretely, the paper proves: if $M$ is a complete connected Riemannian manifold with Ricci curvature bounded below and $\\mu=\\min\\{\\lambda_{\\mathrm{ess}}(M),\\lambda_0(\\widetilde{M})\\}>0$, then for every $\\Lambda<\\mu$ an $n$-sheeted covering of $M$ chosen uniformly at random has, with probability tending to one as $n\\to\\infty$, no Laplacian eigenvalues in $[0,\\Lambda]$ beyond the eigenvalues it inherits from $M$, provided the fundamental group satisfies the representation-theoretic condition (PRP), known for free groups and for fundamental groups of closed orientable surfaces. This extends the 'uniform spectral gap for random covers' phenomenon, previously proved only for hyperbolic surfaces, to every Ricci-bounded manifold whose essential spectrum and universal-cover spectrum have a positive gap. The route is a parametrix built from the resolvent kernel of the universal cover, controlled by the Cheng-Yau gradient estimate instead of an explicit formula. A direct corollary is that every surface of finite type with negative Euler characteristic (orientable if closed), with curvature bounded below and $\\lambda_0(\\widetilde{S})>0$, has random finite covers that are spectrally stable below $\\lambda_0(\\widetilde{S})$.","feed_headline":"Random finite covers gain no new eigenvalues below the spectral gap","feed_subtitle":"The result reaches finite-type surfaces with positive bottom spectrum and hyperbolic manifolds in higher dimensions.","key_machinery":"The load-bearing object is a parametrix $M_\\varphi(\\lambda)$ acting on the new part of the cover, an approximate resolvent with $(\\Delta_{M_\\varphi}-\\lambda)M_\\varphi(\\lambda)=1+T_\\varphi(\\lambda)$ and $\\|T_\\varphi(\\lambda)\\|<1$; invertibility of $1+T_\\varphi(\\lambda)$ makes $\\Delta-\\lambda$ surjective, hence, by self-adjointness, injective on the new subspace, so no new eigenvalue can lie in $[0,\\Lambda]$. The parametrix is the sum of an end piece, built from the Cheeger-Colding cutoff (Theorem 2.12) and the Dirichlet resolvent of $M\\setminus K$ using $\\lambda_{\\mathrm{ess}}(M)>0$, and an interior piece built from the universal-cover resolvent kernel $G_\\lambda(x,y)=\\int_0^\\infty e^{\\lambda t}p_t(x,y)\\,dt$ truncated by a $\\Gamma$-equivariant cutoff $\\chi_{T,y}$: $R_T(\\lambda,x,y)=\\chi_{T,y}(x)G_\\lambda(x,y)$. Off the diagonal its error kernel $L_T(\\lambda,x,y)=(\\Delta_x-\\lambda)R_T(\\lambda,x,y)$ is controlled by Proposition 4.5, $|L_T(\\lambda,x,y)|\\le (C/T)G_\\lambda(x,y)$, which follows from the Cheng-Yau gradient estimate applied to the positive eigenfunction $G_\\lambda(\\cdot,y)$ for $d(x,y)\\ge 1/2$, with $C$ independent of $\\lambda$ and $T$. $\\lambda$-Lipschitz bounds on $G_\\lambda$ and $\\nabla_xG_\\lambda$ (Lemma 5.1 and Proposition 5.2), again from the gradient estimate applied to the difference quotient $F=(G_{\\lambda_2}-G_{\\lambda_1})/(\\lambda_2-\\lambda_1)$, let the interval $[0,\\Lambda]$ be reduced to a finite grid. At each grid point, finite-dimensional truncation of the Hilbert-Schmidt pieces turns the error norm into the operator inequality (1.2) between a sum $\\sum a(\\gamma)\\otimes(\\rho^0_n\\circ\\varphi)(\\gamma)$ on $\\mathbb{C}^r\\otimes V^0_n$ and the regular-representation sum on $\\mathbb{C}^r\\otimes\\ell^2(\\Gamma)$; the finitely supported maps $a_i$ in Theorem A are these truncations, and (PRP)/(PRS) assert that the inequality holds for random, respectively for some, permutation representations.","core_discovery":"The central claim, on the paper's own terms, is Theorem A (proved as Theorem 7.6 in the greater generality of a finite intermediate covering of an infinite normal covering $\\hat p:\\hat M\\to M$): spectral stability below $\\mu$ is a finite, explicit condition on permutation representations of the deck group $\\Gamma$. For any $0<\\Lambda<\\mu$ and $\\varepsilon>0$, there are finitely many finitely supported matrix-valued maps $a_i:\\Gamma\\to\\mathrm{Mat}_{r_i\\times r_i}(\\mathbb{C})$ such that every finite cover $M_\\varphi$, attached as $\\Gamma\\backslash(\\hat M\\times\\{1,\\dots,n\\})$ to a homomorphism $\\varphi:\\Gamma\\to S_n$, is automatically $[0,\\Lambda]$-stable whenever $\\varphi$ satisfies the operator-norm inequality (1.2) with each $a_i$: the norm of $\\sum_\\gamma a_i(\\gamma)\\otimes(\\rho^0_n\\circ\\varphi)(\\gamma)$ on $\\mathbb{C}^{r_i}\\otimes V^0_n$ is at most the norm of the same sum against the regular representation on $\\mathbb{C}^{r_i}\\otimes\\ell^2(\\Gamma)$, up to $\\varepsilon$. The Laplacian on the 'new' subspace $L^2_{\\mathrm{new}}(M_\\varphi)$ then has no spectrum in $[0,\\Lambda]$, which is exactly $[0,\\Lambda]$-stability. Theorem B follows because (PRP) guarantees the inequality holds asymptotically almost surely for uniform random $\\varphi$, and (PRP) is known for finitely generated free groups and for closed orientable surface groups. The new analytic input is that the universal-cover resolvent kernel $G_\\lambda(x,y)=\\int_0^\\infty e^{\\lambda t}p_t(x,y)\\,dt$ needs no explicit formula: a uniform gradient bound, supplied by the Cheng-Yau estimate, suffices, so the result is independent of any special structure of $\\hat M$ beyond its positive spectral bottom.","pith_inferences":["The proof gives no rate for the probability that a random cover is unstable; a natural quantitative follow-up is a concentration or large-deviation bound showing that this probability decays exponentially in the covering degree for closed hyperbolic surfaces, going beyond the qualitative almost-sure statement.","The mechanism suggests the same bottom-of-spectrum rigidity for other operators with a positive Green's kernel and Cheng-Yau-type gradient control, notably Schrodinger operators with a potential or Hodge Laplacians on forms under curvature assumptions, where random covers should likewise show no new low eigenvalues.","The almost-sure statement is plausibly the optimal generic picture at the bottom: classical constructions produce specific covers of closed hyperbolic surfaces with eigenvalues arbitrarily close to zero, so unstable covers must be exponentially rare exceptions rather than the generic case, a contrast the present methods do not quantify.","Conceptually, the paper casts the Riemannian random-cover problem as the metric analogue of random lifts of graphs, with the regular representation governing the new spectrum and the universal cover playing the role of the infinite tree; the new point is that no tree-like structure is needed, only a positive spectral bottom, which one could test by simulating random covers of compact hyperbolic 3-"],"forward_implications":["Corollary C: for any surface $S$ of finite type with negative Euler characteristic, orientable if closed, with a complete metric of curvature bounded below and $\\lambda_0(\\widetilde{S})>0$, finite coverings of $S$ are asymptotically almost surely $[0,\\Lambda]$-stable for every $\\Lambda<\\lambda_0(\\widetilde{S})$; in particular closed hyperbolic surfaces are covered for every $\\Lambda<1/4$.","The results reach hyperbolic manifolds outside the surface world: random finite covers of Schottky manifolds and of quasi-Fuchsian threefolds of type one are asymptotically almost surely stable below the universal-cover bottom (Remark F).","When the fundamental group satisfies (PRS) instead, as limit groups do, the manifold admits a sequence of stable finite coverings of degree tending to infinity, and in fact a tower of them (Theorems D and G).","Because $\\lambda_0$ is an eigenvalue of multiplicity equal to the number of connected components, the paper notes that $[0,\\Lambda]$-stability with $\\Lambda\\ge\\lambda_0(M)$ forces the covering to be connected, so the random covers of Theorem B are almost surely connected."],"supporting_citations":[{"why":"Supplies the Cheng-Yau gradient estimate (Theorem 2.13), the analytic input behind the uniform bound on the resolvent-kernel gradient (Proposition 4.5) that replaces the explicit resolvent formula of the hyperbolic setting.","marker":"[14]"},{"why":"Supplies the strategy the paper adapts: decomposition into interior and ends, finite-dimensional approximation of the interior parametrix, and parametrization of finite covers by permutation representations.","marker":"[15]"},{"why":"Supplies the cutoff functions (Theorem 2.12) used, together with the Dirichlet resolvent of the complement of a compact set, to build the end parametrix from the assumption that the essential spectrum is positive.","marker":"[12]"},{"why":"Recent proof that fundamental groups of closed orientable surfaces satisfy (PRP); this is the hypothesis behind Theorem B and Corollary C in the closed-surface case.","marker":"[22]"},{"why":"Proof that finitely generated free groups satisfy (PRP), covering the non-compact surface cases and applications to Schottky-type manifolds.","marker":"[6]"},{"why":"Proof that limit groups satisfy (PRS), the input for the existence of sequences and towers of stable coverings (Theorems D, E and Corollary G).","marker":"[17]"},{"why":"Brooks' theorem: for a closed manifold the universal cover has positive spectral bottom exactly when the fundamental group is non-amenable; this gives the positive bottom in the closed case.","marker":"[9]"},{"why":"Theorem 2.4, showing that a non-amenable covering strictly raises the bottom of the spectrum when the base has an isolated bottom, used to verify the positivity assumption for surfaces in Corollaries C and E.","marker":"[24]"}],"fun_headline_variants":["Random finite covers resist new eigenvalues below spectral gap","Finite covers stay spectrally stable below the gap","Explicit condition rules out new eigenvalues on finite covers","Spectral gap stability for random finite covers","No new eigenvalues on finite covers: a finite check"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hangs on one analytic estimate (Proposition 4.5): away from the diagonal, the gradient of the Green's function of the Laplacian on the universal cover is bounded by a fixed multiple of the Green's function itself, with the multiple independent of the energy parameter; if that multiple grows without bound as the energy approaches the bottom of the universal-cover spectrum, the parametrix norm bounds collapse and this proof would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Random finite covers resist new eigenvalues below spectral gap","Finite covers stay spectrally stable below the gap","Explicit condition rules out new eigenvalues on finite covers","Spectral gap stability for random finite covers","No new eigenvalues on finite covers: a finite check"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000433,"raw_usage":{"total_tokens":2239,"prompt_tokens":1011,"completion_tokens":1228,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":1155}},"tokens_in":627,"tokens_out":1228,"duration_ms":10422,"temperature":1.0,"reasoning_tokens":1155,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:48:24.143230+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On real hyperbolic space $\\mathbb{H}^m$ the Green's function is explicit, so Proposition 4.5 can be checked numerically: compute the supremum of $\\|\\nabla_x G_\\lambda(x,y)\\|/G_\\lambda(x,y)$ over $d(x,y)\\ge 1/2$ and $\\lambda\\in[0,1/4-\\delta]$, and verify that it is a finite constant depending only on $m$, the Ricci bound, and $1/4$; a divergence as $\\lambda\\uparrow 1/4$ would falsify the estimate and break the proof. To test the theorem itself, one would compute spectra of random covers of a fixed closed hyperbolic 3-manifold (whose fundamental group is free, hence (PRP) holds) for a fixed $\\Lambda<1/4$: if a new eigenvalue in $[0,\\Lambda]$ appears with probability bounded away from zero as the covering degree grows, Theorem B is false.","supporting_citations":[{"cited_title":"Cheng and S.T","cited_arxiv_id":null,"evidence_quote":"Supplies the Cheng-Yau gradient estimate (Theorem 2.13), the analytic input behind the uniform bound on the resolvent-kernel gradient (Proposition 4.5) that replaces the explicit resolvent formula of the hyperbolic setting."},{"cited_title":"Hide and M","cited_arxiv_id":null,"evidence_quote":"Supplies the strategy the paper adapts: decomposition into interior and ends, finite-dimensional approximation of the interior parametrix, and parametrization of finite covers by permutation representations."},{"cited_title":"Cheeger and T","cited_arxiv_id":null,"evidence_quote":"Supplies the cutoff functions (Theorem 2.12) used, together with the Dirichlet resolvent of the complement of a compact set, to build the end parametrix from the assumption that the essential spectrum is positive."},{"cited_title":"Bordenave and B","cited_arxiv_id":null,"evidence_quote":"Proof that finitely generated free groups satisfy (PRP), covering the non-compact surface cases and applications to Schottky-type manifolds."},{"cited_title":"Louder and M","cited_arxiv_id":null,"evidence_quote":"Proof that limit groups satisfy (PRS), the input for the existence of sequences and towers of stable coverings (Theorems D, E and Corollary G)."},{"cited_title":"Brooks, The fundamental group and the spectrum of the Laplacian","cited_arxiv_id":null,"evidence_quote":"Brooks' theorem: for a closed manifold the universal cover has positive spectral bottom exactly when the fundamental group is non-amenable; this gives the positive bottom in the closed case."},{"cited_title":"Polymerakis, Coverings preserving the bottom of the spectrum.J","cited_arxiv_id":null,"evidence_quote":"Theorem 2.4, showing that a non-amenable covering strictly raises the bottom of the spectrum when the base has an isolated bottom, used to verify the positivity assumption for surfaces in Corollaries C and E."}],"review_version":1}