Totally disconnected locally compact groups

DefinitionCaprace's class \(\mathcal{S}\)#✓fully proved

A group \(G\) with a topology belongs to the class \(\mathcal{S}\) when it is a topological group that is Hausdorff, totally disconnected, locally compact and not discrete, is generated as a group by some compact subset, and is topologically simple: \(G\) is nontrivial and its only closed normal subgroups are the trivial subgroup and \(G\) itself.

def IsInClassS (G : Type) [Group G] [TopologicalSpace G] : Prop :=
  IsTopologicalGroup G ∧ T2Space G ∧ TotallyDisconnectedSpace G ∧ LocallyCompactSpace G ∧
    ¬ DiscreteTopology G ∧ (∃ K : Set G, IsCompact K ∧ Subgroup.closure K = ⊤) ∧ Nontrivial G ∧
    ∀ N : Subgroup G, N.Normal → IsClosed (N : Set G) → N = ⊥ ∨ N = ⊤

Used by 1

Used byCaprace's problem on local isomorphism classes (Kourovka 19.22)

DefinitionLocally isomorphic topological groups#✓fully proved

Two topological groups are locally isomorphic: they contain open subgroups that are isomorphic as topological groups.

def LocallyIsomorphic (G H : Type) [Group G] [TopologicalSpace G] [Group H] [TopologicalSpace H] :
    Prop :=
  ∃ (U : OpenSubgroup G) (V : OpenSubgroup H), Nonempty ((U : Subgroup G) ≃ₜ* (V : Subgroup H))

Used by 1

Used byCaprace's problem on local isomorphism classes (Kourovka 19.22)

DefinitionCaprace's problem on local isomorphism classes (Kourovka 19.22)#✓fully proved

The statement that the groups in Caprace's class \(\mathcal{S}\) fall into uncountably many local isomorphism classes: there is a family \((G_i)_{i \in I}\) of groups in \(\mathcal{S}\), indexed by an uncountable set \(I\), such that \(G_i\) and \(G_j\) are not locally isomorphic whenever \(i \ne j\).

def uncountablyManyLocalIsomorphismClasses : Prop :=
  ∃ (ι : Type) (_ : Uncountable ι) (G : ι → Type) (_ : ∀ i, Group (G i))
    (_ : ∀ i, TopologicalSpace (G i)),
    (∀ i, IsInClassS (G i)) ∧ ∀ i j, i ≠ j → ¬ LocallyIsomorphic (G i) (G j)

Uses 2 · Used by 1

Statement usesCaprace's class \(\mathcal{S}\) · Locally isomorphic topological groups

Used byUncountably many local isomorphism classes of simple tdlc groups

TheoremUncountably many local isomorphism classes of simple tdlc groups#●ready to prove

There is a family \((G_i)_{i \in I}\) of groups in Caprace's class \(\mathcal{S}\) (non-discrete, compactly generated, topologically simple, totally disconnected, locally compact Hausdorff topological groups), indexed by an uncountable set \(I\), such that \(G_i\) and \(G_j\) are not locally isomorphic whenever \(i \ne j\). This answers Kourovka Notebook Problem 19.22 (P.-E. Caprace) in the affirmative.

public theorem_wanted uncountablyManyLocalIsomorphismClasses_holds :
    uncountablyManyLocalIsomorphismClasses

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 19.22 (P.-E. Caprace); resolution: I. Castellano, J. Fariña-Asategui, M. E. Garciarena, B. Marchionna, M. Quick, C. D. Reid, S. M. Smith, S. Tornier, M. Vannacci and J. S. Wilson, Uncountably many local isomorphism types of compactly generated simple groups, preprint, arXiv:2609.17410 (2026), Theorem A: there exist 2^ℵ₀ many non-discrete compactly generated simple tdlc groups which are not pairwise locally isomorphic; the Kourovka Notebook (21st issue) marks 19.22 solved: Yes, it is, https://arxiv.org/abs/2609.17410.

Uses 1

Statement usesCaprace's problem on local isomorphism classes (Kourovka 19.22)

3 fully proved1 ready to prove

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