TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Thurston stability: groups of orientation-preserving C¹ interval germs are locally indicable

Statement

Let $G$ be a nontrivial finitely generated subgroup of ${Diff}_{0}^{1, +} (R)$ , the group of germs at $0$ of $C^{1}$ orientation-preserving local diffeomorphisms of $R$ fixing $0$ (C¹ germs of local diffeomorphisms at a point, C¹ germs of local diffeomorphisms form a group). Then there is a surjective homomorphism $G ↠ Z$ ; that is, ${Diff}_{0}^{1, +} (R)$ is locally indicable.

Facts & Assumptions

Given: A nontrivial finitely generated subgroup $G \leq {Diff}_{0}^{1, +} (R)$ with a finite generating set $g_{1}, \dots, g_{m}$ , and representatives of these germs defined near $0$ and fixing $0$ .

[F1]

${Diff}_{0}^{1, +} (R)$ is a group under composition of germs, its elements are germs of $C^{1}$ local diffeomorphisms with positive derivative at $0$ , and a subgroup is a subset containing the identity and closed under products and inverses (C¹ germs of local diffeomorphisms form a group, Group and abelian group, Subgroup).

[F2]

A $C^{1}$ local diffeomorphism of $R$ fixing $0$ with derivative $1$ at $0$ can be written near $0$ as $g (x) = x + y (g) (x)$ with $y (g) (0) = 0$ and $y (g)^{'} (0) = 0$ ; the derivative of a $C^{1}$ map is continuous, so for every $ϵ > 0$ there is a neighbourhood of $0$ on which $∣ y (g)^{'} ∣ < ϵ$ (Continuously differentiable maps, local inverses, and local diffeomorphisms, C¹ germs of local diffeomorphisms at a point).

[F3]

The image of a finitely generated group under a homomorphism is finitely generated (Images of finitely generated and of finite groups are finitely generated and finite).

[F4]

Every finitely generated abelian group is isomorphic to $Z^{r} \oplus$ (finite torsion) for a unique $r \geq 0$ ; a nonzero finitely generated torsion-free abelian group therefore has $r \geq 1$ and admits a surjection onto $Z$ (The fundamental theorem of finitely generated abelian groups from PID modules).

Proof

technique · direct, following Calegari's proof of Theorem 2.119

1.1F1F2F3F4

(The derivative homomorphism.) For a germ $g \in G$ choose a representative and set $d (g) : = \log g^{'} (0)$ ; the value is well defined because representatives agree near $0$ and the derivative at $0$ is a germ invariant, and $d (g h) = d (g) + d (h)$ by the chain rule, so $d : G \to R$ is a homomorphism into the additive group of the reals [F1]. If $d (G) \neq {0}$ , then $d (G)$ is a nonzero finitely generated subgroup of $R$ by [F3], hence torsion-free, and [F4] shows $d (G) ≅ Z^{r}$ with $r \geq 1$ ; projecting onto one free coordinate gives a surjective homomorphism $G ↠ Z$ , and the theorem is proved. Henceforth assume $d (G) = {0}$ , that is, every element of $G$ has derivative $1$ at $0$ .

2.1F2step 1.1

(Normalized displacements.) Shrink a common domain so that every generator is defined and satisfies [F2]; then $g_{j} (x) = x + y (g_{j}) (x)$ with $y (g_{j})^{'} (0) = 0$ . Since $G$ is nontrivial, some generator is not the identity germ, so the open set ${x : w (x) > 0}$ , where $w (x) : = \max_{j} ∣ y (g_{j}) (x) ∣$ , accumulates at $0$ . Fix an enumeration of the rationals and, for every $n \geq 1$ , let $x_{n}$ be the rational of least index lying in the nonempty open set ${x : ∣ x ∣ < 1 / n, w (x) > 0}$ ; then $x_{n} \to 0$ and $w_{n} : = w (x_{n}) > 0$ , and this selection is canonical, so no choice principle is used. The vectors $a_{n} : = (y (g_{j}) (x_{n}) / w_{n})_{j = 1}^{m}$ lie in the compact cube $[- 1, 1]^{m}$ and have maximum norm $1$ ; passing to a convergent subsequence, write $a = (a_{1}, \dots, a_{m})$ for its limit, so $\max_{j} ∣ a_{j} ∣ = 1$ and $a \neq 0$ .

3.1F2step 2.1

(Word estimates.) Fix a word $w$ in the letters $g_{j}^{\pm 1}$ and let $e_{j} (w) \in Z$ be the signed exponent sum of the letter $g_{j}$ in $w$ . We claim that the displacement of the corresponding element, as a function of $x$ , satisfies $y (w) (x_{n}) = w_{n} \sum_{j = 1}^{m} e_{j} (w) a_{j} + o (w_{n}) .$ This follows by induction on the length of $w$ from two estimates: (i) for generators, $y (g_{j}) (x_{n}) = w_{n} a_{j} + o (w_{n})$ by step 2.1; (ii) the composition formula $g \circ h (x) = x + y (h) (x) + y (g) (x + y (h) (x))$ and the continuity of $y (g)^{'}$ with $y (g)^{'} (0) = 0$ give $y (g) (x + y (h) (x)) = y (g) (x) + o (∣ y (h) (x) ∣)$ , so composing adds the displacements up to $o (w_{n})$ uniformly over words whose letters are taken from the fixed finite set, because every partial displacement is $O (w_{n})$ by the induction hypothesis and the increment is taken at points $x_{n} + O (w_{n}) \to 0$ . For an inverse letter, applying the same composition formula to $g^{- 1} \circ g$ at $x_{n}$ gives $y (g^{- 1}) (x_{n}) = - y (g) (x_{n}) + o (w_{n})$ . Multiplying these estimates through the word proves the displayed formula.

4.1F1step 3.1

(The limiting homomorphism.) Define $v (g) : = \lim_{n} y (w) (x_{n}) / w_{n}$ for any word $w$ representing $g \in G$ . The value is independent of the chosen word: if $w, w^{'}$ represent the same germ, then the displacement function of $w^{'} w^{- 1}$ vanishes identically near $0$ , since $w^{'} w^{- 1}$ is the identity germ, while step 3.1 applied to the word $w^{'} w^{- 1}$ gives $\sum_{j} e_{j} (w^{'} w^{- 1}) a_{j}$ as its normalized limit; hence the two normalized limits agree. Moreover $v (g h) = v (g) + v (h)$ , because concatenating representatives concatenates words and signed exponent sums are additive; and $v$ is nontrivial because $v (g_{j}) = a_{j}$ with $\max_{j} ∣ a_{j} ∣ = 1$ [F1]. Thus $v : G \to R$ is a nonzero homomorphism.

5.1F3F4step 1.1step 4.1∎

(Surjection onto $Z$ .) The image $v (G)$ is a nonzero finitely generated subgroup of $R$ by [F3], so it is torsion-free and [F4] identifies it with $Z^{r}$ for some $r \geq 1$ ; projecting onto one free coordinate gives a surjective homomorphism $G ↠ Z$ . Since every nontrivial finitely generated subgroup $G$ was handled in one of the two cases, ${Diff}_{0}^{1, +} (R)$ is locally indicable.

Depends on

Used by

Reeb-Thurston stability for codimension-one leaves with vanishing first real cohomology Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Danny Calegari, Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs; complete author-hosted PDF) (standard reference, not scraped)