False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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.

FALSE: any two infinite finitely generated groups are quasi-isometric

Statement refuted

any two infinite finitely generated groups are quasi-isometric.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

A finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).

[L1]

The word metric of $G$ with respect to $S$ is $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S}$ (The word metric of a group with respect to a generating set).

[L2]

Balls of a word metric are finite if and only if the generating set is finite (Balls of a word metric are finite if and only if the generating set is finite).

[L3]

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L4]

$B (x, r)$ is the open ball, $\overset{ˉ}{B} (x, r)$ the closed ball and $S (x, r)$ the sphere of centre $x$ and radius $r$ . The radius is always a strictly positive real; a ball of radius $0$ or of negative radius is never written in this library. (Open ball, closed ball and sphere in a metric space).

[L5]

A free group on a set $X$ is a group $F (X)$ together with a map $i : X \to F (X)$ such that, for every group $G$ and every function $u : X \to G$ , there is a unique group homomorphism $\hat{u} : F (X) \to G$ satisfying (Free group on a set of generators).

[L6]

A set $A$ is finite when $A \approx n$ for some $n \in N$ . (The cardinality $∣ A ∣$ of a finite set).

[L7]

A map is $(L, C)$ -coarse Lipschitz when $d_{Y} (f (x), f (x^{'})) \leq L d_{X} (x, x^{'}) + C$ , and an $(L, C)$ -quasi-isometric embedding satisfies in addition $L^{- 1} d_{X} (x, x^{'}) - C \leq d_{Y} (f (x), f (x^{'}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

[L8]

In a free group with respect to a free basis, word length is reduced-word length (With respect to a free basis, the word length of an element is the length of its reduced word).

[L9]

For every real number there is a larger natural number (Every complete ordered field is Archimedean).

Refutation

technique · contradiction

1.1

The claim asserts a single quasi-isometry class for all infinite finitely generated groups.

F1assume-contra

1.2

For each integer $n \geq 1$ , the open ball of radius $n + 1$ in the integers has $2 n + 1$ elements. In the free group $F_{2} = ⟨ a, b ⟩$ , the $2^{n}$ positive words of length $n$ in the letters $a, b$ are distinct reduced words, so the corresponding ball has at least $2^{n}$ elements.

L1L2L4L5L6L8

2.1

Suppose there were a quasi-isometry $f : F_{2} \to Z$ . By [L3] choose a coarse Lipschitz quasi-inverse $g : Z \to F_{2}$ , coarse-Lipschitz constants $A, B \geq 0$ for $f$ , and a bound $R \geq 0$ with $d_{F_{2}} (g (f (x)), x) \leq R$ for every $x \in F_{2}$ . If $f (x) = f (x^{'})$ , then $d_{F_{2}} (x, x^{'}) \leq d_{F_{2}} (x, g (f (x))) + d_{F_{2}} (g (f (x^{'})), x^{'}) \leq 2 R$ , so every fibre $f^{- 1} (y)$ lies in the open ball $B_{F_{2}} (g (y), 2 R + 1)$ . By [L1], left multiplication by $g (y)^{- 1}$ bijects that ball with $B_{F_{2}} (e, 2 R + 1)$ , so every fibre has at most $K : = ∣ B_{F_{2}} (e, 2 R + 1) ∣ < \infty$ elements by [L2]. Moreover, if $x \in B_{F_{2}} (e, n + 1)$ then $d_{Z} (f (x), f (e)) \leq A d_{F_{2}} (x, e) + B < A (n + 1) + B$ , so $f (B_{F_{2}} (e, n + 1))$ has at most $2 ⌈ A (n + 1) + B ⌉ + 1$ elements. Hence $2^{n} \leq ∣ B_{F_{2}} (e, n + 1) ∣ \leq K (2 ⌈ A (n + 1) + B ⌉ + 1)$ . But $2^{n} \geq n^{2}$ for $n \geq 4$ by induction, while $2 ⌈ A (n + 1) + B ⌉ + 1 \leq 2 A n + 2 A + 2 B + 3$ . By [L9] choose a natural $n \geq 4$ so large that $n > 4 K A$ and $n^{2} > 2 K (2 A + 2 B + 3)$ ; then $n^{2} > K (2 A n + 2 A + 2 B + 3)$ , contradicting the displayed bound. Thus $F_{2}$ and $Z$ are not quasi-isometric.

L1L2L3L7L8L9step 1.1step 1.2discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

43 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)