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0dd3af4 | 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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.CentralCovers.FinitelyManyTranslationRelations
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section FiniteAlignedConfiguration
variable {α β G : Type*} [Group G] [Finite β] (f : α → G)
theorem finite_table_subgroup_inputs_eventually (L : ℕ)
(K : Subgroup (BoundedRelationCover L f)) {T : Type*} [Group T] [Finite T]
(φ : T →* G) (hφ : ∀ s, φ s ∈ K.map (coverMap L f))
(input : β → T) (v : β → FreeGroup α)
(hv : ∀ b, FreeGroup.lift f (v b) = φ (input b)) :
∃ B : ℕ, ∃ hLB : L ≤ B, ∀ M : ℕ, ∀ hBM : B ≤ M,
∃ ρ : T →* BoundedRelationCover M f,
(coverMap M f).comp ρ = φ ∧
ρ.range ≤ K.map (coverTransition f (le_trans hLB hBM)) ∧
∀ b, ρ (input b) = PresentedGroup.mk (shortRelations M f) (v b) := by
choose y hy hyq using hφ
choose w hw using fun s => PresentedGroup.mk_surjective (shortRelations L f) (y s)
have hwp : ∀ s, FreeGroup.lift f (w s) = φ s := by
intro s
rw [← coverMap_mk L f,hw s,hyq s]
obtain ⟨B₁,hB₁⟩ := finite_table_lift_eventually f φ w hwp
obtain ⟨B₂,hB₂⟩ := finite_word_equalities_eventually f (fun b => w (input b)) v
(fun b => (hwp (input b)).trans (hv b).symm)
refine ⟨L+B₁+B₂,by omega,fun M hM => ?_⟩
obtain ⟨ρ,hρ,hρw⟩ := hB₁ M (by omega)
refine ⟨ρ,hρ,?_,?_⟩
· rintro z ⟨s,rfl⟩
refine ⟨y s,hy s,?_⟩
rw [← hw s,coverTransition_mk,hρw s]
· intro b
rw [hρw]
exact hB₂ M (by omega) b
end FiniteAlignedConfiguration
end SimpleAmenable
end
end
end OAI
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