| import OAI.GroupTheory.UniversalGroup.UniverseTransport | |
| namespace OAI | |
| open Set Metric Topology BigOperators | |
| open Set Metric Topology | |
| open Set Topology | |
| open Set Metric Topology | |
| open scoped NNReal | |
| open Set Metric Topology | |
| open scoped NNReal ENNReal | |
| open Set Metric Topology | |
| open scoped TensorProduct | |
| open Set Metric Topology | |
| namespace UniversalFInfinity | |
| noncomputable section | |
| universe u | |
| theorem universal_group_of_type_FInfinity : | |
| ∃ (H : Type u) (groupH : Group H), | |
| @HasTypeFInfinity H groupH ∧ | |
| ∀ (G : Type u) [Group G] [Group.IsFinitelyPresented G], | |
| ∃ φ : G →* H, Function.Injective φ := by | |
| obtain ⟨U,gU,fpU,i,hi⟩ := Higman.NormalWords.embeds_presentation | |
| RecursiveUniversalThree.enumerator RecursiveUniversalThree.enumerator_primrec.to_comp | |
| let : Group U := gU | |
| let : Group.IsFinitelyPresented U := fpU | |
| have hU : ConcreteEnvelopes.IsUniversal U := by | |
| intro V _ _ | |
| obtain ⟨j,hj⟩ := RecursiveUniversalThree.universal (G := V) | |
| exact ⟨i.comp j,hi.comp hj⟩ | |
| obtain ⟨k,hk,hk_inj⟩ := ConcreteEnvelopes.self_maps U hU | |
| let : Fact (Function.Injective (k (.inl ConcreteEnvelopes.P₂.identity))) := ⟨hk_inj⟩ | |
| let : TopologicalSpace U := ⊥ | |
| let : DiscreteTopology U := ⟨rfl⟩ | |
| let f := k (.inl ConcreteEnvelopes.P₂.identity) | |
| let H₀ := AscendingTorus.Torus f | |
| let : TopologicalSpace H₀ := ⊥ | |
| let : DiscreteTopology H₀ := ⟨rfl⟩ | |
| refine ⟨ULift.{u} H₀,inferInstance,?_,?_⟩ | |
| · exact Brown.geometric_lifted_of_uniform | |
| (TorusGeometry.diagram_milnor_uniform_nullhomotopy k hk) | |
| · intro G _ _ | |
| obtain ⟨j,hj⟩ := RecursiveUniversalThree.universal (G := G) | |
| refine ⟨MulEquiv.ulift.symm.toMonoidHom.comp ((AscendingTorus.of f).comp (i.comp j)),?_⟩ | |
| exact MulEquiv.ulift.symm.injective.comp ((AscendingTorus.of_injective f hk_inj).comp | |
| (hi.comp hj)) | |
| end | |
| end UniversalFInfinity | |
| end OAI | |