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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphTopology.GraphNeighborhoodCompact
import OAI.Probability.CriticalPercolation.GraphGeometry.NormalOrbitAction
import OAI.Probability.CriticalPercolation.GraphGeometry.LocallyConnectedActionKernel
namespace OAI
section
noncomputable section
open Set Function Topology
open scoped Classical
namespace CriticalPercolation.GraphTopology
variable {X : Type*} (Γ : SimpleGraph X)
def orbitKernel (H : Subgroup (Γ ≃g Γ)) [H.Normal] : Subgroup (Γ ≃g Γ) :=
(NormalOrbit.blockHom (X := X) H).ker
instance orbitKernel_normal (H : Subgroup (Γ ≃g Γ)) [H.Normal] :
(orbitKernel Γ H).Normal := MonoidHom.normal_ker _
lemma mem_orbitKernel_iff (H : Subgroup (Γ ≃g Γ)) [H.Normal] (g : Γ ≃g Γ) :
g∈orbitKernel Γ H ↔ ∀ x, g x∈MulAction.orbit H x := by
constructor
· intro hg x
have hh := congrArg (fun p : Equiv.Perm (NormalOrbit.Blocks (X := X) H) =>
p (NormalOrbit.block H x)) hg
exact (NormalOrbit.block_eq_iff H (g x) x).mp hh
· intro hg
apply Equiv.ext
intro b
induction b using Quotient.inductionOn with
| h x => exact (NormalOrbit.block_eq_iff H (g x) x).mpr (hg x)
lemma le_orbitKernel (H : Subgroup (Γ ≃g Γ)) [H.Normal] : H≤orbitKernel Γ H :=
NormalOrbit.le_ker_blockHom H
variable [TopologicalSpace X] [DiscreteTopology X]
local instance : TopologicalSpace (Γ ≃g Γ) := autTopology Γ
lemma orbitKernel_closed (H : Subgroup (Γ ≃g Γ)) [H.Normal] :
IsClosed (orbitKernel Γ H : Set (Γ ≃g Γ)) := by
have he : (orbitKernel Γ H : Set (Γ ≃g Γ))=
⋂ x : X, {g : Γ ≃g Γ | g x∈MulAction.orbit H x} := by
ext g
simp only [Set.mem_iInter,Set.mem_ofPred_eq,SetLike.mem_coe,mem_orbitKernel_iff]
rw [he]
exact isClosed_iInter (fun x => isClosed_eval_mem Γ x _)
lemma compact_orbitKernel (H : Subgroup (Γ ≃g Γ)) [H.Normal]
(hH : IsCompact (H : Set (Γ ≃g Γ))) :
IsCompact (orbitKernel Γ H : Set (Γ ≃g Γ)) := by
have hfin : ∀ x : X, (MulAction.orbit H x).Finite := by
intro x
have hh := compact_finite_vertex_images Γ (H : Set (Γ ≃g Γ)) hH x
apply hh.subset
rintro y ⟨h,rfl⟩
exact ⟨h.val,h.property,rfl⟩
apply (compact_coordinate_bounds Γ (MulAction.orbit H) hfin).of_isClosed_subset
(orbitKernel_closed Γ H)
intro g hg
exact ⟨(mem_orbitKernel_iff Γ H g).mp hg,
(mem_orbitKernel_iff Γ H g⁻¹).mp ((orbitKernel Γ H).inv_mem hg)⟩
lemma open_orbitKernel_of_locallyConnected_quotient
(H : Subgroup (Γ ≃g Γ)) [H.Normal]
[LocallyConnectedSpace ((Γ ≃g Γ) ⧸ H)] :
IsOpen (orbitKernel Γ H : Set (Γ ≃g Γ)) := by
let B := NormalOrbit.Blocks (X := X) H
let _ : TopologicalSpace B := ⊥
let _ : DiscreteTopology B := ⟨rfl⟩
have hcont : ∀ x : X, Continuous (fun q : (Γ ≃g Γ) ⧸ H =>
q • (NormalOrbit.block H x : B)) := by
intro x
apply (QuotientGroup.isQuotientMap_mk H).continuous_iff.mpr
change Continuous (fun g : Γ ≃g Γ => NormalOrbit.block H (g x))
exact (continuous_of_discreteTopology : Continuous (NormalOrbit.block (X := X) H)).comp
(continuous_eval Γ x)
have hsub : (QuotientGroup.mk' H) ⁻¹'
connectedComponent (1 : (Γ ≃g Γ) ⧸ H) ⊆ (orbitKernel Γ H : Set (Γ ≃g Γ)) := by
intro g hg
apply (mem_orbitKernel_iff Γ H g).mpr
intro x
have hh := ((isPreconnected_connectedComponent (x := (1 : (Γ ≃g Γ) ⧸ H))).image
(fun q : (Γ ≃g Γ) ⧸ H => q • (NormalOrbit.block H x : B))
(hcont x).continuousOn).subsingleton
have he := hh ⟨QuotientGroup.mk' H g,hg,rfl⟩ ⟨1,mem_connectedComponent,rfl⟩
apply (NormalOrbit.block_eq_iff H (g x) x).mp
simpa only [NormalOrbit.mk_smul_block,one_smul,RelIso.smul_def] using he
apply (orbitKernel Γ H).isOpen_of_mem_nhds (g := 1)
apply Filter.mem_of_superset _ hsub
apply (isOpen_connectedComponent.preimage QuotientGroup.continuous_mk).mem_nhds
change (QuotientGroup.mk' H) 1∈connectedComponent (1 : (Γ ≃g Γ) ⧸ H)
rw [map_one]
exact mem_connectedComponent
theorem compact_open_orbitKernel_of_locallyConnected_quotient
(H : Subgroup (Γ ≃g Γ)) [H.Normal]
(hH : IsCompact (H : Set (Γ ≃g Γ)))
[LocallyConnectedSpace ((Γ ≃g Γ) ⧸ H)] :
IsCompact (orbitKernel Γ H : Set (Γ ≃g Γ)) ∧
IsOpen (orbitKernel Γ H : Set (Γ ≃g Γ)) :=
⟨compact_orbitKernel Γ H hH,open_orbitKernel_of_locallyConnected_quotient Γ H⟩
theorem compact_open_orbitKernel_of_open_locallyConnected_subquotient
(H L : Subgroup (Γ ≃g Γ)) [H.Normal]
(hH : IsCompact (H : Set (Γ ≃g Γ))) (hL : IsOpen (L : Set (Γ ≃g Γ)))
[LocallyConnectedSpace (L ⧸ H.subgroupOf L)] :
IsCompact (orbitKernel Γ H : Set (Γ ≃g Γ)) ∧
IsOpen (orbitKernel Γ H : Set (Γ ≃g Γ)) := by
refine ⟨compact_orbitKernel Γ H hH,?_⟩
let B := NormalOrbit.Blocks (X := X) H
let _ : TopologicalSpace B := ⊥
let _ : DiscreteTopology B := ⟨rfl⟩
apply ConnectedAction.open_kernel_of_open_locallyConnected_subquotient
(NormalOrbit.blockHom (X := X) H) _ L H hL (le_orbitKernel Γ H)
intro b
induction b using Quotient.inductionOn with
| h x =>
change Continuous (fun g : Γ ≃g Γ => NormalOrbit.block H (g x))
exact (continuous_of_discreteTopology : Continuous (NormalOrbit.block (X := X) H)).comp
(continuous_eval Γ x)
end CriticalPercolation.GraphTopology
end
end
end OAI
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