False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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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.
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FALSE: cobounded and cocompact are identical without extra hypotheses

Statement

Cobounded and cocompact are identical without extra hypotheses.

Facts & Assumptions

Given: The trivial action of the trivial group on the open interval $(0, 1)$ with its usual metric. Here cocompact means that the orbit space of the action is compact.

[L1]

Coboundedness means that some bounded subset has orbit-union equal to the whole space (Isometric, proper, and cobounded actions on metric spaces).

[L2]

The absolute-value metric makes $R$ a metric space, hence its open interval $(0, 1)$ inherits the usual metric (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).

Refutation

technique · direct

1.1L1L2

The interval $(0, 1)$ is bounded, so for the trivial action its single orbit already covers the space. Thus the action is cobounded by [L1].

1.2L2algebra

The orbit space is again $(0, 1)$ , which is not compact: the open cover $U_{n} : = (0, 1 - 1 / n)$ for $n \geq 2$ covers it, but every finite subfamily misses points sufficiently close to $1$ . So the action is not cocompact.

2.1step 1.1step 1.2∎

Step 1.1 gives coboundedness while step 1.2 denies cocompactness, refuting the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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. Löh, Geometric Group Theory, Sections 4.4 and 5.1 (standard reference, not scraped)