DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-29

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 Markov property of finitely presented groups

Definition

A class property $P$ of finitely presented groups is a Markov property if there exist finitely presented groups $G_{+}$ and $G_{-}$ such that:

$G_{+}$ has property $P$ ,
$G_{-}$ does not have property $P$ , and
$G_{-}$ cannot be embedded into any finitely presented group having property $P$ .

This is the exact hypothesis pattern used in the Adian-Rabin theorem.

Depends on

The isomorphism problem for a class of finite presentations

Used by

Adian-Rabin: every Markov property is undecidable on finite presentations Remark

Dependency tree · two levels

2 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

Alex Bishop, Minicourse: On Decision Problems in Groups (standard reference, not scraped)
MacTutor History of Mathematics, Word problems (standard reference, not scraped)