Published paper
A proof of Selberg's conjecture
MSC 2020
22E4622E3022E4022E1522C05
Abstract
The paper proves the Kazhdan–Margulis uniform conjugation theorem and a uniform positive lower bound for covolumes of discrete subgroups, then applies them to Selberg's conjecture. The uniform local theorem is the proof-critical input in the Garland–Raghunathan rank-one fundamental-domain argument.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statement:
- Theorem 1 and corollary · MathNet exact English version of record, physical PDF pp. 2–4 (printed pp. 148–150)Uniform bounded conjugation of sufficiently small elements in arbitrary discrete subgroups, and the resulting positive lower bound for quotient covolume.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Version of record · Mathematics of the USSR-Sbornik 4 (1968), no. 1, 147–152
D. A. Kazhdan and G. A. Margulis. A proof of Selberg's conjecture. Mathematics of the USSR-Sbornik 4 (1968), no. 1, 147–152.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The uniform conjugation theorem and its positive lower-covolume corollary are correctly stated and are the Kazhdan–Margulis inputs used in the Garland–Raghunathan rank-one reduction argument.
Theorem 1No issue found
A uniformly bounded conjugation expands every sufficiently small element of a discrete subgroup
Exact MathNet English version of record, physical PDF p. 2 (printed p. 148)
The theorem supplies one neighborhood and group-dependent constants, independent of the discrete subgroup, together with a bounded conjugating element that expands every subgroup element lying in the neighborhood. This is the uniform local input invoked by Garland and Raghunathan to control cusp rays.
Corollary to Theorem 1No issue found
Finite-volume quotients have a uniform positive covolume lower bound
Exact MathNet English version of record, physical PDF p. 2 (printed p. 148)
Iterating Theorem 1 removes nonidentity elements from a fixed smaller identity neighborhood; that neighborhood then injects into the quotient and gives the claimed uniform Haar-measure lower bound.
02Proofs6 reported findingsFurther proof verification needed
The proof's mathematical argument closes after an explicit standard fiber-dimension repair, but its source record is not fully correct: the cited Rosenlicht 1956 article contains no theorem matching the displayed semicontinuity claim, and the exact proof text of Borel 1956 remained access-restricted. The Borel–Tits containment is exact, Borel's structural statements are independently verified, and Hartshorne supplies the missing semicontinuity theorem, so these source defects do not overturn Theorem 1.
Lemma 2No issue found in the reviewed proof
The subgroup generated in a sufficiently small canonical-coordinate neighborhood is nilpotent
Exact MathNet English version of record, physical PDF p. 3 (printed p. 149)
Successive commutators contract toward the identity while remaining in the discrete subgroup, so the local subgroup is nilpotent. Passing successively to connected central subgroups gives the stated connected nilpotent envelope.
Lemma 3Correct after disclosed source repair
Every connected nilpotent subgroup is expanded by some semisimple conjugation
Exact MathNet English version of record, physical PDF pp. 3–4 (printed pp. 149–150)
After taking the algebraic closure, the proof splits the nilpotent group into toral and unipotent parts. Ordered maximal tori and the standard fiber-dimension theorem produce a torus whose nonpositive weight space misses the nilpotent Lie algebra, yielding uniform local expansion. Borel–Tits supplies the exact positive-root containment. The printed Rosenlicht citation does not supply the semicontinuity theorem, so that step is closed explicitly by Hartshorne rather than silently attributed to Rosenlicht.
Assembly of Theorem 1No issue found in the reviewed proof
Compactness makes the nilpotent-subgroup expansion uniform
Exact MathNet English version of record, physical PDF p. 4 (printed p. 150)
The family of connected nilpotent subalgebras sits in a compact Grassmannian model. Lemmas 2 and 3 therefore admit one finite uniform choice of the neighborhood and constants, which is exactly the assertion of Theorem 1.
Recursive source routeDependency scope verified with one adverse citation finding
The exact classical article inputs and the source repair are represented separately
Exact MathNet English version of record, physical PDF pp. 2–4 (printed pp. 148–150), citations [9]–[11]
Pontryagin supplies canonical Lie-group coordinates. Borel–Tits 1965 supplies the exact positive-root containment; Borel 1956 supplies the nilpotent direct product, with its statement independently checked because the original scan was gated. The complete Rosenlicht 1956 article contains no matching semicontinuity theorem, and Hartshorne supplies the standard result that repairs the proof. The later Selberg-conjecture argument is outside the Garland–Raghunathan edge.
Borel–Tits citation [10]Exact historical source verified
The positive-root containment is exact and applicable
Borel–Tits exact Numdam VOR, physical PDF pp. 11–12, 38–39, and 70–71 (printed pp. 64–65, 91–92, and 123–124)
Sections 2.3 and 4.15–4.16 and Lemma 8.3/Proposition 8.4 place the connected unipotent radical inside the positive-root unipotent group associated with a suitable ordered maximal torus. This is the precise assertion used after the direct-product decomposition; the graph does not over-assign that preceding decomposition to Borel–Tits.
Rosenlicht citation [11]Source attribution not independently verified
The cited article does not contain the claimed semicontinuity theorem
Rosenlicht exact issue scan, complete article physical PDF pp. 181–223 / printed pp. 401–443; Kazhdan–Margulis physical PDF p. 4 / printed p. 150
An exhaustive source check found no theorem or proof about semicontinuity of dimensions in an algebraic family. The nearest Proposition 1 is a fixed homogeneous-space dimension formula and does not imply the displayed local inequality. The inequality itself is standard and correct by the fiber-dimension theorem, which is therefore shown as a separate repair edge.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.