arXiv:math/0002184v1
Limit distributions of expanding translates of certain orbits on homogeneous spaces
math.RTmath.DG22E40
Abstract
Let be a Lie group and a lattice in . Suppose is a non-compact simple Lie group realized as a Lie subgroup of , and the image of on is dense. Let be a diagonalizable element of not contained in a compact subgroup, and let be its expanding horospherical subgroup. The paper proves density and measure-equidistribution results for expanding translates of open subsets of , and derives applications to equivariant factors and unipotent flows on finite-volume homogeneous spaces.
Dependence graphs
Proof lineage
The Dirichlet spectrum
A statement-restricted proof-dependence graph for arXiv:2412.05858v1. It covers Theorems 1–5, Corollary 1, and Lemmas 1–4; recursively follows every proof-critical non-book result; and stops only at disclosed published books or edited/seminar volumes. Corollary 1 is verified only for finite positive critical limsup: its endpoint assertions at zero and infinity do not follow from Theorem 3 and formula (5). The Dani 1981 branch retains the Garland–Raghunathan article and Kazhdan–Margulis input. The latter is expanded through the exact Borel–Tits, Borel, and Rosenlicht articles, preserving an insufficient Rosenlicht citation alongside its Hartshorne repair, and recursively through the Weil and Lang–Weil article chain rather than replacing historical sources with retrospective books.
Open dependence graph →AI-generated audit
Audit summary
Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.
Current report
Detailed mathematical audit
01Statements2 reported findingsNo statement issues found
The quantitative return alternative, the linearization theorem near a proper homogeneous stratum, and the elementary projection lemma used by Kleinbock–Weiss are all correctly stated in the generality required downstream.
Theorem 2.2No issue found
The recurrence alternative covers arbitrary lattices
arXiv:math/0002184v1 PDF pages 5–7
After reducing compact normal and solvable factors, the rank-one case is supplied by Dani's cusp argument and the higher-rank case by arithmeticity followed by the Eskin–Mozes–Shah theorem. These cases exhaust the semisimple reduction used in the statement.
Theorem 4.1 and Lemma 5.1No issue found
Linearization and the unipotent projection estimate have the required uniformity
arXiv:math/0002184v1 PDF pages 8–10
Theorem 4.1 gives the precise small-neighborhood alternative imported by Kleinbock–Weiss. Lemma 5.1 is an explicit finite polynomial-expansion argument and yields a norm bound uniform over the fixed unipotent box.
02Proofs3 reported findingsNo proof issues found
The selected proofs close after the explicit rank-one, higher-rank, and linearization imports. Two historical preprint locators have different numbering in the final published articles; the graph records the exact final theorem numbers.
Proof of Theorem 2.2No issue found in the reviewed proof
The rank split is complete
arXiv:math/0002184v1 PDF pages 6–7
The structural reductions preserve compactness of the quotient fibers. Rank one uses Dani's Lemma 2.4 route; higher rank uses Margulis arithmeticity to reach the arithmetic setting of Eskin–Mozes–Shah. Raghunathan's book supplies the lattice and compact-factor reductions.
Published-number mappingSource locator clarified
The EMS preprint theorem was renumbered in the final article
arXiv:math/0002184v1 · Theorem 2.2 proof; EMS final Section 3
The paper cites the then-preprint EMS Theorem 3.6. In the 1997 GAFA version, the combined quantitative non-divergence result is Theorem 3.9. The statement is the same operative result, not a missing theorem.
Proof of Theorem 4.1No issue found in the reviewed proof
The cited polynomial-trajectory linearization theorem supplies the conclusion
arXiv:math/0002184v1 PDF page 8
Shah 1994 Proposition 5.4 and Theorem 5.2 give the neighborhood alternative. The polynomial-map formulation in the present paper is a direct specialization of that result.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.