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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 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.FiniteSubspaceOrbit
import OAI.Probability.CriticalPercolation.LinearGroups.LocalFurstenbergBound
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
variable {K V G : Type*} [Field K] [AddCommGroup V] [Module K V] [Group G]
def FullSpan (ρ : G →* (V ≃ₗ[K] V)) : Prop :=
Submodule.span K (Set.range (fun g => (ρ g).toLinearMap))=⊤
def StrongFullSpan (ρ : G →* (V ≃ₗ[K] V)) : Prop :=
∀ H : Subgroup G, H.FiniteIndex → FullSpan (ρ.comp H.subtype)
lemma submodule_eq_bot_or_top_of_fullSpan (ρ : G →* (V ≃ₗ[K] V)) (hfull : FullSpan ρ)
(W : Submodule K V) (hinv : ∀ g, W.map (ρ g).toLinearMap=W) : W=⊥ ∨ W=⊤ := by
by_cases hW : W=⊥
· exact Or.inl hW
right
obtain ⟨w,hw,hw0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hW
obtain ⟨φ,hφ⟩ := Projective.exists_dual_eq_one K hw0
apply top_unique
intro y _
let T := φ.smulRight y
have hT : T∈Submodule.span K (Set.range (fun g => (ρ g).toLinearMap)) := hfull ▸ Submodule.mem_top
have hm : ∀ A∈Submodule.span K (Set.range (fun g => (ρ g).toLinearMap)), A w∈W := by
intro A hA
induction hA using Submodule.span_induction with
| mem A hA =>
obtain ⟨g,rfl⟩ := hA
rw [←hinv g]
exact Submodule.mem_map.mpr ⟨w,hw,rfl⟩
| zero => exact W.zero_mem
| add A B hA hB ihA ihB => exact W.add_mem ihA ihB
| smul c A hA ihA => exact W.smul_mem c ihA
simpa only [T,LinearMap.smulRight_apply,hφ,one_smul] using hm T hT
lemma no_finite_subspace_orbit_of_strongFullSpan (ρ : G →* (V ≃ₗ[K] V))
(hρ : StrongFullSpan ρ) (W : Submodule K V) (hW0 : W≠⊥) (hW1 : W≠⊤) :
¬(Set.range (fun g => W.map (ρ g).toLinearMap)).Finite := by
intro hf
obtain ⟨H,hHi,hH⟩ := finiteIndex_subspace_stabilizer ρ W hf
exact (submodule_eq_bot_or_top_of_fullSpan (ρ.comp H.subtype) (hρ H hHi) W hH).elim hW0 hW1
end CriticalPercolation.LinearGrowth
namespace CriticalPercolation.LinearGrowth
variable {K V G : Type*} [NontriviallyNormedField K] [NormedAddCommGroup V]
variable [NormedSpace K V] [ProperSpace K] [FiniteDimensional K V] [Group G]
def continuousRepresentation (ρ : G →* (V ≃ₗ[K] V)) : G →* (V ≃L[K] V) where
toFun g := (ρ g).toContinuousLinearEquiv
map_one' := by ext x; change ρ 1 x=x; rw [map_one]; rfl
map_mul' g h := by ext x; change ρ (g*h) x=ρ g (ρ h x); rw [map_mul]; rfl
lemma projective_bound_of_strongFullSpan [Nontrivial V] [DecidableEq G]
(ρ : G →* (V ≃ₗ[K] V)) (hρ : StrongFullSpan ρ)
(S : Finset G) (h1 : (1:G)∈S) (hgen : Subgroup.closure (S:Set G)=⊤)
(d : ℕ) (hg : SparseGrowth.HasDegree S d) :
∃ δ : ℝ, 0<δ ∧ ∀ g : G, ∀ x : V,
δ*‖(continuousRepresentation ρ g).toContinuousLinearMap‖*‖x‖≤‖ρ g x‖ := by
rcases LocalProjective.sparse_local_furstenberg_alternative (continuousRepresentation ρ)
S h1 hgen d hg with horb | hb
· obtain ⟨W,hW0,hW1,hW⟩ := horb
exact False.elim (no_finite_subspace_orbit_of_strongFullSpan ρ hρ W hW0 hW1 hW)
· exact hb
end CriticalPercolation.LinearGrowth
end
end
end OAI
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