| import Mathlib |
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| namespace OAI |
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| /-! Properness and cocompactness transferred through equivariant maps. -/ |
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| section |
| section |
| section |
| noncomputable section |
| noncomputable section |
| open Set Filter Topology |
| open scoped OnePoint |
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| noncomputable section |
| open Set Filter Topology |
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| namespace CannonActions |
| open Set |
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| variable {Γ X Y : Type*} [Group Γ] [TopologicalSpace X] [TopologicalSpace Y] |
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| def Proper (A : Γ →* (X ≃ₜ X)) : Prop := |
| ∀ K : Set X, IsCompact K → {g : Γ | ∃ x ∈ K, A g x ∈ K}.Finite |
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| def Cocompact (A : Γ →* (X ≃ₜ X)) : Prop := |
| ∃ K : Set X, IsCompact K ∧ ∀ x : X, ∃ g : Γ, ∃ y ∈ K, A g y=x |
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| theorem proper_descends {A : Γ →* (X ≃ₜ X)} {B : Γ →* (Y ≃ₜ Y)} |
| {p : X → Y} (hp : IsProperMap p) (hs : Function.Surjective p) |
| (he : ∀ g x, p (A g x)=B g (p x)) (hA : Proper A) : Proper B := by |
| intro K hK |
| apply (hA (p ⁻¹' K) (hp.isCompact_preimage hK)).subset |
| rintro g ⟨y,hy,hgy⟩ |
| obtain ⟨x,rfl⟩ := hs y |
| exact ⟨x,hy,by simpa only [mem_preimage,he] using hgy⟩ |
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| theorem cocompact_descends {A : Γ →* (X ≃ₜ X)} {B : Γ →* (Y ≃ₜ Y)} |
| {p : X → Y} (hp : Continuous p) (hs : Function.Surjective p) |
| (he : ∀ g x, p (A g x)=B g (p x)) (hA : Cocompact A) : Cocompact B := by |
| obtain ⟨K,hK,hAK⟩ := hA |
| refine ⟨p '' K,hK.image hp,?_⟩ |
| intro y |
| obtain ⟨x,rfl⟩ := hs y |
| obtain ⟨g,z,hz,rfl⟩ := hAK x |
| exact ⟨g,p z,mem_image_of_mem p hz,(he g z).symm⟩ |
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| theorem proper_finite_kernel [Nonempty X] {A : Γ →* (X ≃ₜ X)} (hA : Proper A) : |
| (A.ker : Set Γ).Finite := by |
| obtain ⟨x⟩ := ‹Nonempty X› |
| apply (hA {x} isCompact_singleton).subset |
| intro g hg |
| have hh : A g=1 := hg |
| exact ⟨x,mem_singleton x,by simp only [hh,Homeomorph.one_apply,mem_singleton_iff]⟩ |
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| def conjugate (ψ : X ≃ₜ Y) (A : Γ →* (X ≃ₜ X)) : Γ →* (Y ≃ₜ Y) where |
| toFun g := (ψ.symm.trans (A g)).trans ψ |
| map_one' := by |
| ext y |
| simp only [map_one,Homeomorph.trans_apply,Homeomorph.one_apply,Homeomorph.apply_symm_apply] |
| map_mul' g h := by |
| ext y |
| simp only [map_mul,Homeomorph.trans_apply,Homeomorph.mul_apply,Homeomorph.symm_apply_apply] |
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| lemma conjugate_equivariant (ψ : X ≃ₜ Y) (A : Γ →* (X ≃ₜ X)) (g : Γ) (x : X) : |
| ψ (A g x)=conjugate ψ A g (ψ x) := by |
| change ψ (A g x)=ψ (A g (ψ.symm (ψ x))) |
| rw [ψ.symm_apply_apply] |
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| theorem proper_conjugate (ψ : X ≃ₜ Y) {A : Γ →* (X ≃ₜ X)} (hA : Proper A) : |
| Proper (conjugate ψ A) := |
| proper_descends ψ.isProperMap ψ.surjective (conjugate_equivariant ψ A) hA |
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| theorem cocompact_conjugate (ψ : X ≃ₜ Y) {A : Γ →* (X ≃ₜ X)} (hA : Cocompact A) : |
| Cocompact (conjugate ψ A) := |
| cocompact_descends ψ.continuous ψ.surjective (conjugate_equivariant ψ A) hA |
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| end CannonActions |
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| namespace CannonActions |
| open Set |
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| variable {X Y Z Γ : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [Group Γ] |
| abbrev OrderedTriples (X : Type*) := {t : Fin 3 → X // Function.Injective t} |
| def triplesMap (f : X ≃ₜ Y) (t : OrderedTriples X) : OrderedTriples Y := |
| ⟨f ∘ t.val,f.injective.comp t.property⟩ |
| lemma continuous_triplesMap (f : X ≃ₜ Y) : Continuous (triplesMap f) := by |
| apply Continuous.subtype_mk |
| apply continuous_pi |
| intro i |
| exact f.continuous.comp ((continuous_apply i).comp continuous_subtype_val) |
| def triplesCongr (f : X ≃ₜ Y) : OrderedTriples X ≃ₜ OrderedTriples Y where |
| toFun := triplesMap f |
| invFun := triplesMap f.symm |
| left_inv t := by apply Subtype.ext; funext i; exact f.symm_apply_apply _ |
| right_inv t := by apply Subtype.ext; funext i; exact f.apply_symm_apply _ |
| continuous_toFun := continuous_triplesMap f |
| continuous_invFun := continuous_triplesMap f.symm |
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| def triplesHom : (X ≃ₜ X) →* (OrderedTriples X ≃ₜ OrderedTriples X) where |
| toFun := triplesCongr |
| map_one' := by apply Homeomorph.ext; intro t; rfl |
| map_mul' f g := by apply Homeomorph.ext; intro t; rfl |
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| def triplesAction (A : Γ →* (X ≃ₜ X)) : Γ →* (OrderedTriples X ≃ₜ OrderedTriples X) := triplesHom.comp A |
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| lemma triples_conjugate (A : Γ →* (X ≃ₜ X)) (ψ : X ≃ₜ Y) : |
| triplesAction (conjugate ψ A)=conjugate (triplesCongr ψ) (triplesAction A) := by |
| apply MonoidHom.ext |
| intro g |
| apply Homeomorph.ext |
| intro t |
| rfl |
| end CannonActions |
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| end |
| end |
| end |
| end |
| end |
| end |
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| end OAI |
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