ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-28

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.

Regular Cayley trees of free groups

Example

If $F_{r}$ is a free group with free basis $S$ of size $r$ , then the undirected Cayley graph of $(F_{r}, S)$ is a tree and every vertex has valence $2 r$ . Orienting each geometric edge in both directions turns it into a simplicial tree.

Facts & Assumptions

Given: A free group $F_{r}$ with free basis $S$ .

[L1]

The Cayley graph of a free group with respect to a free basis is a tree. (The Cayley graph of a free group with respect to a free basis is a tree)

[L2]

On finite pieces, the simplicial-tree notion matches the published tree notion. (For finite graphs, the simplicial-tree notion agrees with the published finite-tree notion)

Verification

technique · direct

1.1L1given

By [L1], the underlying simple Cayley graph is a tree. Each vertex has one edge labelled by each basis element and by its inverse, so its valence is $2 r$ .

2.1L1L2step 1.1algebra∎

Replacing every geometric edge by the two corresponding orientations does not create a cycle; it only records both directions explicitly. Thus the same graph becomes a simplicial tree, in agreement with the finite bridge principle [L2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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) (standard reference, not scraped)