Length functions on groups and rigidity

Shengkui Ye

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:55 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves exactly the base case the model claimed was likely open: if H carries a purely positive length function, any homomorphism from a higher–Q-rank arithmetic group Γ has finite image (Theorem 8.2), via the key Heisenberg-vanishing lemma (Lemma 5.2) and an induction along a subnormal series (Lemma 8.3). This yields the full virtually poly-positive target result (Theorem 0.3). The argument that Γ contains a Heisenberg subgroup is supplied using rank-2 Q-root subsystems (A2, B2, G2). Hence the statement is established in the paper, contradicting the model’s “likely open” assessment.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript develops a general and elegant rigidity framework using length functions, proving strong finiteness results for homomorphisms from higher-rank arithmetic groups to broad target classes. The key Heisenberg center vanishing argument is both novel and effective. A few details about normality/finite-index propagation in the inductive step could be spelled out for clarity, but the argument is standard and persuasive.