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CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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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.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Hyperbolic groups have solvable word problem

Statement

Every hyperbolic group has solvable word problem.

Facts & Assumptions

Given: A hyperbolic group G with a finite Dehn presentation ⟨S∣R⟩.

[L1]

In a Dehn presentation, every nonempty freely reduced trivial word contains a subword longer than half of a relator (Hyperbolic groups admit finite Dehn presentations).

[A1]

Replacing such a long subword by the complementary shorter subword strictly decreases word length and preserves the represented group element.

Proof

technique · direct
1.1

Starting from any input word w, repeatedly apply the replacement from [A1] whenever [L1] finds a long relator half. Because length strictly decreases, the process terminates after finitely many steps.

L1A1
2.1

If the algorithm stops at the empty word, then w=1 in G. Conversely, if w=1 in G and the current reduced word is nonempty, [L1] says that another shortening move exists, so the procedure cannot terminate early. Thus the algorithm decides whether w represents the identity.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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