Published paper
Abstract
Theorem 0.6 supplies the finite-cusp reduction data and separation properties used to prove the compactness and short-vector assertions in Kleinbock–Weiss Proposition 3.1.
Role in dependence graphs
Proof-critical source
Bounded trajectories of quasi-rays
This paper is included only for the following marked statement:
- Theorem 0.6 · source theorem 0.6; use in Kleinbock–Weiss pp. 8–9Supplies finite-cusp reduction data, the cusp-lattice property, and cusp-overlap separation used to prove parts (2)–(3) of Kleinbock–Weiss Proposition 3.1.
Proof-critical source
Submanifold-genericity of -actions and uniform multiplicative Diophantine approximation
This paper is included only for the following marked statement:
- Theorem 0.6 · source Theorem 0.6; use in Kleinbock–Weiss pp. 8–9Finite-cusp reduction, cusp lattices, and overlap separation.
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statement:
- Theorem 0.6 · Annals of Mathematics 92 (1970), Theorem 0.6; exact PNAS announcement Theorem 1.2, physical scans pp. 309–313; exact Dani 1981 restatement, printed pp. 365–366Finite rank-one cusp representatives, lattice intersections in the corresponding horospherical groups, and a finite Siegel-set covering.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact official PNAS version-of-record page-scan fallback · the Annals version of record was not accessed
Howard Garland and M. S. Raghunathan. Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups. Annals of Mathematics 92 (1970), no. 2, 279–326. Target record; reviewed through the exact official PNAS announcement, Fundamental domains for lattices in rank one semisimple Lie groups, Proceedings of the National Academy of Sciences 62 (1969), no. 2, 309–313.
The exact Annals of Mathematics version of record was subscription-gated during this audit. The exact official PMC page scans of the complete five-page 1969 PNAS version-of-record announcement were reviewed together with Dani's exact 1981 consumer restatement. The PNAS source states the marked theorem but gives only an indication of its proof and explicitly defers details elsewhere. This report does not claim inspection of the Annals file, its Theorem 0.6 typography, or its complete proof.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The exact PNAS announcement states the finite-cusp, cusp-lattice, covering, finite-overlap, and deep-cusp separation conclusions of Theorem 1.2 with the required rank-one and nonuniform-lattice hypotheses. Its first two clauses match the exact finite-cusp input restated in Dani 1981 Proposition 3.1. This audit does not claim textual inspection of the gated Annals Theorem 0.6.
PNAS Theorem 1.2(i)–(ii)Correct · focal dependency verified at statement level
A finite family of Siegel cusps covers the quotient and has lattice cusp stabilizers
Exact PNAS version of record · printed page 310 · Theorem 1.2(i)–(ii)
For a nonuniform lattice in a connected linear simple real-rank-one group, the theorem produces a positive cutoff, a relatively compact open subset of the expanding horospherical group, and a finite set . Clause (i) says every conjugated cusp group meets the lattice in a lattice, and clause (ii) says the union of the resulting Siegel sets covers modulo the lattice. These are exactly the finite-cusp and cusp-lattice assertions used by the marked consumer.
Official PMC page scan of PNAS page 310 ↗PNAS Theorem 1.2(iii)–(iv)No issue found
Finite overlap and deep-cusp separation are stated with the needed uniform quantifiers
Exact PNAS version of record · printed page 310 · Theorem 1.2(iii)–(iv)
Clause (iii) makes the set of lattice translates meeting the finite union of Siegel sets finite. Clause (iv) supplies one deeper cutoff that works simultaneously for every lattice element and every pair of cusp representatives, forcing equality of the cusp labels and membership of the resulting conjugate in the parabolic. The uniform order of these quantifiers is explicit in the statement.
Dani 1981 Proposition 3.1Dependency match verified
The consumer's reduction data are an exact reformulation of the selected rank-one clauses
Exact Dani 1981 Springer version of record · printed pages 365–366 · Proposition 3.1 and proof
Dani asks for a finite set of cusp elements, one compact factor in the opposite parabolic, one split-torus cutoff, global coverage, and a lattice intersection with the opposite horospherical group at each cusp. Inverting the PNAS covering and using the opposite Iwasawa convention converts Theorem 1.2(i)–(ii) into this form. Dani's proof explicitly attributes the real-rank-one component to Garland–Raghunathan, separately from Borel's arithmetic reduction theory.
Dani 1981 version of record ↗Access boundaryExplicit theorem-level fallback
The reviewed journal source is the PNAS announcement, not the Annals version of record
Official PNAS pages 309–313; official Annals article record for volume 92 (1970), pages 279–326
The official Annals record confirms the target title, authors, volume, year, pages, and DOI, but its publisher proof text was not publicly accessible in this audit. The exact official PNAS pages contain the complete announcement and the marked theorem statement. They do not authenticate the wording, numbering, or full proof of Annals Theorem 0.6, and this report does not represent them as the Annals version of record.
Official Annals article record ↗02Proofs4 reported findingsFurther proof verification needed
The PNAS source presents a coherent proof architecture for Theorem 1.2 but explicitly calls Section 3 only an indication of the proof, omits the proofs of several essential lemmas, and says that details will appear elsewhere. The selected finite-cusp chain is therefore strongly supported but not completely verifiable from the accessible fallback.
Selected proof chainCoherent outline · suspected incomplete proof
Rays, compact cusp quotients, and finiteness of ray directions give the finite cover
Exact PNAS version of record · printed pages 311–313 · Lemmas 3.2, 3.6, 3.7, and 3.10
The symmetric-space Dirichlet domain is exhausted by rays. Lemma 3.2 makes the horospherical quotient attached to each ray compact, Lemmas 3.6–3.7 show that unit ray directions form a finite set, and Lemma 3.10 assembles finitely many Siegel regions covering the quotient. Conjugating those ray groups yields Theorem 1.2(i), while the assembled union yields clause (ii). However, Lemmas 3.3–3.5 and 3.7 are stated without full proofs, and Lemma 3.10(a) is described only as a fairly straightforward deduction.
Explicit deferral of detailsNot independently verified completely
The announcement does not supply the complete Annals proof
Introduction on printed page 309 and final paragraph on printed page 313 · exact PNAS version of record
The introduction says that Section 3 gives an indication of the proof of Theorem 1.2. The final paragraph says that details of the arguments described there will appear elsewhere. The accessible five-page source therefore supports the theorem statement and proof plan but cannot justify a globally correct proof rating for the inaccessible 48-page Annals article.
Kazhdan–Margulis 1968Proof-critical parent verified in its exact version of record
The Selberg-hypothesis paper is a genuine and audited proof-critical parent
Printed page 309, final paragraph of the introduction; printed page 311, discussion after Lemmas 3.3–3.5; reference 4 on printed page 313
The announcement expressly says that its proof of Theorem 1.2 relies on the results of Kazhdan and Margulis, and later says that Lemmas 3.3 and 3.5 use the ideas of that paper. Those lemmas feed central Lemma 3.2, which is needed for the cusp-lattice clause and then for the finite-cusp assembly. The dependence graph therefore includes a separate Kazhdan–Margulis node, and its uniform local theorem and covolume corollary were checked in the exact MathNet English version of record rather than treated as contextual literature.
Official MathNet record and English translation ↗Unmarked separation branchExternal input for Theorem 1.2(iii), not for selected clauses (i)–(ii)
Borel's carried-over inequality is outside the selected Dani dependency
Exact PNAS version of record · printed page 313 · equations (3.11)–(3.13) and final paragraph
The finite-overlap argument says that the proof of the lower bound in Lemma 3.12 carries over from Borel's cited discussion and that it implies equation (3.11). This is proof-critical for the full Siegel-property clause (iii), but Dani Proposition 3.1 uses only the finite cusp set, coverage, and cusp-lattice properties supplied by clauses (i)–(ii). It does not create an additional parent on the currently marked edge.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.