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

The translation length of a tree automorphism without inversions

Definition

Let $g$ be an automorphism of a simplicial tree $T$ that acts without inversions. Its translation length is

$ℓ (g) : = \min_{v \in V (T)} d_{T} (v, g v),$

using the path metric of The simplicial path metric on a tree.

The existence of this minimum, together with the elliptic/hyperbolic classification, is the content of Tree automorphisms without inversions are either elliptic or hyperbolic ↗, recorded here in justified_by.

Depends on

Used by

Dependency tree · two levels

5 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

Jean-Pierre Serre, Trees (standard reference, not scraped)