Alphabeta Math
RemarkRemark: Literature-sourcedProof: Not suppliedSession-authored (Fable 5 assisted)audited 2026-08-29‡ sources checked 2026-08-29‡ not proved here
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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Stallings's theorem on ends and splittings

Statement

A finitely generated group has more than one end if and only if it splits nontrivially over a finite subgroup, either as an amalgamated free product or as an HNN extension.

Remarks

This theorem lies beyond the present page because ends have not yet been developed. It is recorded here only to mark the later bridge from tree actions to large-scale geometry.

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