ExampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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.

Free groups act geometrically on regular trees

Example

Let $F_{r}$ be a free group of rank $r \geq 2$ , and let $T_{X}$ be its Cayley graph with respect to a free basis $X$ . Give its geometric realization unit-length edges and the induced path metric. Then $T_{X}$ is a regular metric tree, and the left translation action of $F_{r}$ on that tree is geometric. Consequently $F_{r}$ is quasi-isometric to that tree.

Facts & Assumptions

Given: A free basis $X$ of a free group $F_{r}$ with $r \geq 2$ , the geometric realization of its Cayley graph $T_{X}$ with unit-length edges, and the identity vertex $x_{0}$ .

[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]

A geometric action is isometric, proper, and cobounded (Geometric actions on a metric space).

[L3]

Under a geometric action on a geodesic metric space, every orbit map is a quasi-isometry (The Svarc-Milnor lemma).

Verification

technique · direct

1.1L1L2algebra

By [L1], the geometric realization $T_{X}$ is a tree with unit-length edges, so the unique arc between any two points is a geodesic in its path metric. Left translation extends linearly across edges and preserves lengths. If bounded sets $B, C \subseteq T_{X}$ satisfy $g B \cap C \neq \emptyset$ , choose $x \in B$ with $g x \in C$ . Then $d (x_{0}, g x_{0}) \leq d (x_{0}, x) + d (x_{0}, g x)$ , which is bounded by constants depending only on $B, C$ . Finite valence makes the vertex ball of that radius finite, and the free vertex action has a distinct orbit vertex for each $g$ ; hence only finitely many $g$ can carry $B$ into $C$ . Thus the action is proper. The vertex orbit is all vertices, and every point of an edge lies within $1 / 2$ of a vertex, so the action is cobounded. It is geometric by [L2].

2.1L3step 1.1∎

Applying [L3] to the action on the geodesic metric tree in step 1.1 shows that the orbit map from $F_{r}$ into $T_{X}$ is a quasi-isometry.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)