False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: vertex stabilizers are literally the chosen vertex groups without conjugacy ambiguity

Statement

In the Bass-Serre action, every vertex stabilizer is literally equal to one of the chosen vertex groups, with no conjugacy ambiguity.

Facts & Assumptions

Given: The Bass-Serre action of a graph-of-groups fundamental group.

[L1]

The stabilizer of a vertex coset $γ G_{v}$ is $γ G_{v} γ^{- 1}$ . (The fundamental group acts without inversions on its Bass-Serre tree)

[L2]

A one-segment graph of groups has the corresponding amalgamated free product as its fundamental group. (A one-segment graph of groups gives an amalgamated free product)

[L3]

A positive-length normal word in a free product is nonidentity. (Normal form theorem for free products with amalgamation)

Refutation

technique · direct

1.1

Consider the one-segment graph of groups with vertex groups $C_{2}$ and $C_{3}$ and trivial edge group. Its fundamental group is $C_{2} * C_{3}$ by [L2]. Let $a$ and $b$ be nonidentity elements of the two factors. By [L1], the vertex coset $b C_{2}$ has stabilizer $b C_{2} b^{- 1}$ .

L1L2given

2.1

If $b a b^{- 1}$ lay in $C_{2}$ , say $b a b^{- 1} = c$ , then $b a b^{- 1} c^{- 1}$ would be a reduced positive-length free-product word representing the identity, contrary to [L3]. Hence $b C_{2} b^{- 1} \neq C_{2}$ : this vertex stabilizer is a conjugate of, but not literally equal to, the chosen vertex group.

L1L3step 1.1algebra∎

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

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