Local word identities

DefinitionLocal word identities in connected groups (Kourovka 18.71)#✓fully proved

The statement that for every connected Hausdorff topological group \(G\) and every word \(W(x, y)\) in the free group on two generators \(x\), \(y\), if \(W(a, b) = 1\) for all \((a, b)\) in some nonempty open subset of \(G \times G\), then \(W(a, b) = 1\) for all \(a, b \in G\).

The problem is posed for connected topological groups; here they are also taken to be Hausdorff, which weakens the statement.

def localWordIdentityHoldsGlobally : Prop :=
  ∀ (G : Type) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [T2Space G],
    ConnectedSpace G → ∀ (W : FreeGroup (Fin 2)) (U : Set (G × G)), IsOpen U → U.Nonempty →
      (∀ p ∈ U, FreeGroup.lift ![p.1, p.2] W = 1) → ∀ x y : G, FreeGroup.lift ![x, y] W = 1

Used by 1

Used byLocal word identities need not hold globally in connected groups

TheoremLocal word identities need not hold globally in connected groups#●ready to prove

There are a connected Hausdorff topological group \(G\), a word \(W(x, y)\) in the free group on two generators \(x\), \(y\) and a nonempty open subset \(U\) of \(G \times G\) such that \(W(a, b) = 1\) for all \((a, b) \in U\) but \(W(a, b) \ne 1\) for some \(a, b \in G\). This answers Kourovka Notebook Problem 18.71 in the negative.

public theorem_wanted localWordIdentityHoldsGlobally_refuted : ¬ localWordIdentityHoldsGlobally

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.71 (J. Mycielski); resolution: I. N. Zyabrev and E. A. Reznichenko, On identities in connected topological groups, preprint, arXiv:2406.05203 (2024), abstract: a connected (indeed path-connected) Hausdorff topological group in which, for an odd n, the identity x^n = e holds in some neighbourhood of the identity but not in the whole group; the Kourovka Notebook (21st issue) marks 18.71 solved (odd n > 10^10), https://arxiv.org/abs/2406.05203.

Uses 1

Statement usesLocal word identities in connected groups (Kourovka 18.71)

1 fully proved1 ready to prove

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