TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-09

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Both bounds for last-term weighted distortion

Statement

Let $G$ be finitely generated nilpotent of class $c \geq 1$ , and $H = γ_{c} (G)$ . For fixed finite word metrics on $G$ and $H$ , there is $C \geq 1$ with $C^{- 1} ∣ h ∣_{H}^{1 / c} - C \leq ∣ h ∣_{G} \leq C ∣ h ∣_{H}^{1 / c} + C$ for every $h \in H$ . If $H$ is infinite, $Δ (n) = \max {∣ h ∣_{H} : h \in H, ∣ h ∣_{G} \leq n}$ lies between positive multiples of $n^{c}$ for all sufficiently large integers n. If $H$ is finite, $Δ$ is bounded.

Facts & Assumptions

Given: $H = γ_{c}$ is finitely generated abelian and central. All generating sets are fixed.

[F1]

A short word representing an element of the last term has only last-layer coordinates, bounded by a constant times max(1,n)^c (Weighted collection with finite-order carries).

[F2]

Powers of each fixed last-term element have ambient length at most a constant times the c-th root of the exponent (Power compression in the last lower-central term).

[F3]

Write the finitely generated abelian last term as $Z^{r} \oplus F$ with F finite (Integer abelian structure and rank by finite reduction).

[F4]

Word length is subadditive and invariant under inversion (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

Proof

1.1F1F3F4

Choose the decomposition $H = ⟨ z_{1} ⟩ \oplus \dots \oplus ⟨ z_{r} ⟩ \oplus F$ and finite generators for F, using these as the last-layer coordinate system. Every chosen coordinate generator has a fixed finite H-word. For $h \neq 1$ , collect a shortest G-word of length n: all earlier coordinates vanish, each last free exponent is $O (n^{c})$ , and the finitely many residue exponents are bounded. Multiplying fixed H-words for these powers gives $∣ h ∣_{H} \leq A n^{c} + B$ ; the same inequality with max(1,n) handles h=1. Increasing A gives $∣ h ∣_{H} \leq A \max (1, ∣ h ∣_{G})^{c}$ . Taking roots gives the required lower ambient bound with an additive constant.

1.2F2F3F4

For any finite H-generating set V, let L be the maximum absolute free-coordinate entry of a member of $V \cup V^{- 1}$ , enlarged to at least 1. Projection to each free coordinate is additive, so a shortest H-word gives $∣ a_{j} ∣ \leq L ∣ h ∣_{H}$ for $h = z_{1}^{a_{1}} \dots z_{r}^{a_{r}} f$ . Let M be the maximum G-length of an element of the finite set F. Power compression and subadditivity now give $∣ h ∣_{G} \leq \sum_{j} C_{j} ∣ a_{j} ∣^{1 / c} + M \leq (\sum_{j} C_{j}) L^{1 / c} ∣ h ∣_{H}^{1 / c} + M$ , omitting zero exponents. This proves the other pointwise bound.

2.1F2step 1.1step 1.2

For each n the defining maximum for $Δ (n)$ exists: the finite alphabet of G has only finitely many words of length at most n, and the identity belongs to the intersection. Step 1.1 gives $Δ (n) \leq A \max (1, n)^{c}$ . If H is infinite then r>=1. The first coordinate estimate in step 1.2 gives $∣ z_{1}^{m} ∣_{H} \geq ∣ m ∣ / L$ , while F2 gives $∣ z_{1}^{m} ∣_{G} \leq K ∣ m ∣^{1 / c}$ with K>=1. For $m = ⌊ (n / K)^{c} ⌋$ and sufficiently large n, m is at least $(n / K)^{c} / 2$ and at least 1, so $Δ (n) \geq n^{c} / (2 L K^{c})$ .

3.1step 1.1step 1.2step 2.1∎

If H is finite, its H-word lengths have a finite maximum, bounding $Δ$ for all n. Its finite ambient and intrinsic diameters are absorbed by the pointwise additive constants. At n=0 the intersection contains only the identity and $Δ (0) = 0$ . For c=1 the exponent is one and the same argument applies to H=G. Choose one C larger than all constants in the two pointwise estimates.

Source notes

Druţu–Kapovich, Geometric Group Theory (837-page edition), Proposition 14.20 and Lemma 14.21, printed pp.504–508 (finite last terms handled separately locally); Druţu–Kapovich, Lectures on Geometric Group Theory (585-page draft), Corollary 12.39, printed pp.322–323. Revised Proposition 14.20 and draft Corollary 12.39 supply the two routes. The infinite-H hypothesis is necessary for positive-power distortion growth; finite H is handled separately.

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Sources

Druţu–Kapovich, Geometric Group Theory (837-page edition) (standard reference, not scraped)
Druţu–Kapovich, Lectures on Geometric Group Theory (585-page draft) (standard reference, not scraped)