CounterexampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

A single map exhibiting a quasi-isometry that is discontinuous, non-injective and non-surjective

Statement refuted

A quasi-isometry must be continuous, injective, or surjective.

Facts & Assumptions

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

[F1]

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

[L1]

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).

[L2]

It is written $⌊ x ⌋$ and called the integer part, or floor, of $x$ . (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[L3]

Define $d_{R} : R \times R \to R$ by $d_{R} (x, y) : = ∣ x - y ∣$ . (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L4]

(a) $f$ is continuous at every point of $X$ in the $ε$ - $δ$ sense. (Metric continuity characterisations, with countable choice for the sequential converse).

Counterexample

technique · constructive

1.1F1L1L2L3construct

Let $f : R \to R$ be $f (x) = 2 ⌊ x / 2 ⌋$ , whose image is the even integers, and let $ι$ be the inclusion of that image into $R$ . Then $ι$ is coarse Lipschitz, $f (2 m) = 2 m$ for every even integer $2 m$ , and every real $x$ satisfies $∣ x - f (x) ∣ < 2$ ; so $f$ is a quasi-isometry.

2.1L2L4step 1.1discharge-construct∎

It is discontinuous at every even integer, non-injective on each half-open interval $[2 m, 2 m + 2)$ , and misses every odd integer, so all three failures occur in one map.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 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)