2406.19160
Limits of definable families and dilations in nilmanifolds
Ya’acov Peterzil, Sergei Starchenko
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem 6.12 rigorously establishes the two equivalences via nonstandard methods, strong Γ-density, and a compactness argument (Prop. 6.5, Prop. 6.8, Claim 6.13), and is internally consistent with its definitions of nearest co-commutative cosets and Lmax(F) . By contrast, the candidate solution hinges on an incorrect claim that π0(gL) = π0(g LΓ0) (used to conclude π0(gL) = G/Γ0 when LΓ0 = G), which is generally false (one only gets density/closure). This flaw breaks its Part (1) “nonstandard density” argument. Moreover, it over-claims Part (2) by asserting the ‘properness’ of Hausdorff limits for Γ itself or for any finite-index subgroup, contradicting the phenomenon illustrated in Example 6.6(2), where limits are proper only after passing to some finite-index Γ0 .
Referee report (LaTeX)
\textbf{Recommendation:} no revision
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The theorem gives a uniform, model-theoretic criterion for strong convergence of definable families in nilmanifolds and a dual statement ensuring proper limits after passage to a finite-index lattice. The proof strategy is coherent and leverages the paper’s nonstandard and type-theoretic machinery effectively (strong Γ-density, nearest cosets, compactness). Cross-links to abelianization further solidify correctness and scope.