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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteIndexAction
import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitQuotientTransfer
import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalFiniteGeneration
import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalScales
import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def orbitalActionGroup (o : V) (L : ℕ) (Λ : Subgroup G.Aut) :
Subgroup (G.orbitPower o L ≃g G.orbitPower o L) :=
((G.orbitPowerHom o L).comp Λ.subtype).range
def orbitalActionMap (o : V) (L : ℕ) (Λ : Subgroup G.Aut) :
Λ →* G.orbitalActionGroup o L Λ :=
((G.orbitPowerHom o L).comp Λ.subtype).rangeRestrict
lemma orbitalActionMap_surjective (o : V) (L : ℕ) (Λ : Subgroup G.Aut) :
Surjective (G.orbitalActionMap o L Λ) := MonoidHom.rangeRestrict_surjective _
instance orbitalActionGroup_fg (o : V) (L : ℕ) (Λ : Subgroup G.Aut) [Group.FG Λ] :
Group.FG (G.orbitalActionGroup o L Λ) :=
Group.fg_of_surjective (G.orbitalActionMap_surjective o L Λ)
lemma orbitalActionMap_kernel_fixes (o : V) (L : ℕ) (Λ : Subgroup G.Aut)
(φ : (G.orbitalActionMap o L Λ).ker) : φ • o=o := by
have hh := congrArg Subtype.val φ.property
change (G.orbitPowerHom o L φ.val.val) = 1 at hh
let o' : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩
have ho := congrArg (fun f : G.orbitPower o L ≃g G.orbitPower o L => (f o').val) hh
exact ho
lemma orbitalActionMap_finite_kernel_orbits (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (o : V) (L : ℕ) (Λ : Subgroup G.Aut) :
∀ v : V, (MulAction.orbit (G.orbitalActionMap o L Λ).ker v).Finite := by
let _ : MulAction G.Aut E := G.autEdgeAction
apply G.finite_kernel_orbits_from_one_vertex hc hl
(G.subgroup_preservesEnds G.aut_preservesEnds Λ) (G.orbitalActionMap o L Λ) o
apply (Set.finite_singleton o).subset
rintro v ⟨φ,rfl⟩
exact G.orbitalActionMap_kernel_fixes o L Λ φ
lemma orbitalActionGroup_transitive (o : V) (L : ℕ) (Λ : Subgroup G.Aut)
(horbit : MulAction.orbit Λ o=MulAction.orbit G.Aut o)
(u v : MulAction.orbit G.Aut o) :
∃ φ : G.orbitalActionGroup o L Λ, φ.val u=v := by
have hu : u.val∈MulAction.orbit Λ o := by rw [horbit]; exact u.property
have hv : v.val∈MulAction.orbit Λ o := by rw [horbit]; exact v.property
obtain ⟨a,ha⟩ := hu
obtain ⟨b,hb⟩ := hv
refine ⟨G.orbitalActionMap o L Λ (b*a⁻¹),?_⟩
apply Subtype.ext
change (b*a⁻¹) • u.val=v.val
change a • o=u.val at ha
change b • o=v.val at hb
rw [←ha,mul_smul,inv_smul_smul,hb]
theorem sparse_polynomial_faithful_orbital_model [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(o : V) (d : ℕ)
(hpoly : ∀ R : ℕ, ∃ n≥R, ∃ v : V, (G.vertexBall v n).ncard≤n^d) :
∃ C : ℕ, ∃ Λ : Subgroup G.Aut, Group.FG Λ ∧
(G.orbitPower o (2*C+1)).Connected ∧
(MulAction.orbit G.Aut o).Infinite ∧
(∀ u, ((G.orbitPower o (2*C+1)).neighborSet u).Finite) ∧
(∀ u v, ∃ φ : G.orbitalActionGroup o (2*C+1) Λ, φ.val u=v) ∧
(∀ v : V, (MulAction.orbit (G.orbitalActionMap o (2*C+1) Λ).ker v).Finite) ∧
(∃ S : Set V, S.Finite ∧ ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) ∧
(∀ R : ℕ, ∃ n≥R, ∀ u : MulAction.orbit G.Aut o,
{v | ∃ p : (G.orbitPower o (2*C+1)).Walk u v, p.length≤n}.ncard≤n^(d+1)) := by
obtain ⟨C,hC,hdense,hscales⟩ := G.orbital_power_polynomial_scales hc hl hq o d hpoly
obtain ⟨Λ,hΛ,horbit,hreps⟩ := G.finitelyGenerated_quasiTransitive_subgroup hc hl hq o
refine ⟨C,Λ,hΛ,hC,G.dense_set_infinite hl hdense,?_,?_,?_,hreps,hscales⟩
· exact G.powerOn_finite_neighbors hl _ _
· exact G.orbitalActionGroup_transitive o (2*C+1) Λ horbit
· exact G.orbitalActionMap_finite_kernel_orbits hc hl o (2*C+1) Λ
end CriticalPercolation.BondGraph
end
end
end OAI
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