ASYMPTOTIC PROPERTIES OF INFINITESIMAL CHARACTERS AND APPLICATIONS

Andrés Sambarino

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper defines VJv and Vψv and proves Theorem A using an affine-geometry approach: Proposition 8.1 identifies the derivative of the Jordan projection with the a-component of the Margulis invariant for G ⋉ Ad g, and Corollary 8.4 then yields convexity and non-empty interior for VJv; Proposition 9.1 shows full-variation elements are Zariski-dense and their Jordan projections intersect any open subcone of Lρ; Proposition 9.7 establishes convexity of normalized variations Vψv . By contrast, the candidate solution sketches a different route (Schottky families, coarse additivity, Kato perturbation), but it makes unjustified moves: e.g., it claims left/right multiplication by fixed elements does not affect variations in character variety coordinates (it does), and it assumes without proof that full loxodromic variation plus Zariski-density implies that the dλγ(v) span a (needed for non-empty interior). These gaps are addressed in the paper via the affine-limit-cone machinery and Zariski-closure arguments (e.g., Lemma 2.9 for additivity limits and the semigroup-to-group closure step) . Hence the paper’s results are correct and complete, while the model outline is not rigorous enough.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This work introduces and studies the cone of Jordan variations and normalized variations at an infinitesimal level for Zariski-dense representations into semisimple groups, proving convexity, non-empty interior, and strong density properties. The strategy via the affine group G⋉Ad g and Margulis invariants is elegant and powerful, connecting higher-rank dynamics with affine geometry. The results have meaningful applications to pressure forms and higher Teichmüller theory. The exposition is generally clear, though a few cross-references could be tightened and some proofs (e.g., transversality-to-Zariski-closure implications) could benefit from additional guiding remarks.