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.

Fixed subtrees and minimal invariant subtrees

Definition

Let a group $G$ act on a simplicial tree $T$ . For a subgroup $H \leq G$ , the fixed-vertex set is the global fixed-point set $V (T)^{H}$ from The fixed-point sets $X^{g}$ and $X^{G}$ of a group action. When this set is nonempty, the subtree spanned by those fixed vertices is the fixed subtree of $H$ and is denoted $T^{H}$ .

A nonempty subtree $Y \subseteq T$ is $H$ -invariant when $h (Y) = Y$ for every $h \in H$ . A minimal $H$ -invariant subtree is an $H$ -invariant subtree that contains no smaller nonempty $H$ -invariant subtree. It need not exist for an arbitrary action, and when it does exist it need not be unique in general.

Depends on

Used by

The nonempty fixed-vertex set of a tree automorphism is a subtree Lemma

Dependency tree · two levels

6 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)