OAI/Topology/ArtinGroups/ParabolicIntersections/VisualTrivialSegment.lean

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2 declarations, 0 contain sorry, 0 declared as axioms

The visual tree has a segment with trivial pointwise stabilizer. This proves the segment step in CMMW (2026), Theorem 3.2, using the unconditional parabolic intersection theorem and rank descent.

  1. lemma OAI.HarmonicArtin.ParabolicIntersections.visual_pair_isParabolic line 19
  2. theorem OAI.HarmonicArtin.ParabolicIntersections.visual_exists_trivial_pair line 28

    Source documentation

    A finite segment of the genuine Bass--Serre tree has trivial stabilizer.

    theorem visual_exists_trivial_pair (hIrred : IsIrreducible M) :
        ∃ u v : VisualVertex M Y Z, u≠v ∧ pairStabilizer (G := Artin M) u v=⊥ := by
    …

    Mentions: OAI.HarmonicArtin.ParabolicIntersections.IsIrreducible, OAI.HarmonicArtin.ParabolicIntersections.VisualVertex, OAI.HarmonicArtin.Artin