Orbit-blocking words

DefinitionKourovka Problem 20.88 (statement)#✓fully proved

The statement that for every \(r \ge 1\) and every element \(u\) of the free group \(F_r\) on \(r\) generators there is \(v \in F_r\) such that, for every automorphism \(\varphi\) of \(F_r\) for which the reduced word of \(\varphi(u)\) is cyclically reduced, the reduced word of \(v\) is not a subword (a contiguous factor) of the reduced word of \(\varphi(u)\).

def existsOrbitBlockingWord : Prop :=
  ∀ r : ℕ, 1 ≤ r → ∀ u : FreeGroup (Fin r), ∃ v : FreeGroup (Fin r),
    ∀ φ : MulAut (FreeGroup (Fin r)), FreeGroup.IsCyclicallyReduced (φ u).toWord →
      ¬ v.toWord <:+: (φ u).toWord

Used by 1

Used byOrbit-blocking words exist (Kourovka Problem 20.88)

TheoremOrbit-blocking words exist (Kourovka Problem 20.88)#●ready to prove

For every \(r \ge 1\) and every element \(u\) of the free group \(F_r\) on \(r\) generators there is \(v \in F_r\) such that, for every automorphism \(\varphi\) of \(F_r\) for which the reduced word of \(\varphi(u)\) is cyclically reduced, the reduced word of \(v\) is not a subword (a contiguous factor) of the reduced word of \(\varphi(u)\).

public theorem_wanted existsOrbitBlockingWord_holds : existsOrbitBlockingWord

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.88 (V. Shpilrain); resolution: L. Koch-Hyde, S. O'Connor, É. Olive and V. Shpilrain, Orbit-blocking words in free groups, preprint, arXiv:2505.00477 (2025), Theorem 6.1 and Section 6: for every w ∈ F_r there are w-orbit-blocking words, i.e. words v that are not subwords of the cyclic reduction of φ(w) for any automorphism φ (built from primitivity-blocking words such as x₁² ⋯ x_r² x₁); Kourovka Notebook (21st issue), Problem 20.88: Yes, it is true, https://arxiv.org/abs/2505.00477.

Uses 1

Statement usesKourovka Problem 20.88 (statement)

1 fully proved1 ready to prove

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