| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.AlignedFiniteTables |
| import OAI.GroupTheory.SimpleAmenable.Configurations.FiniteAlignedConfiguration |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FullSectorCompatibility |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ControlCalculus |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.CoordinateFamilyCompare |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.UniformTangentTransport |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PolygonAtomLaw |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.AxisSplitting |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.SectorDifferenceRange |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.AnchorConstants |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GrowthGeneration |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ClippedLineSlideAction |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GlobalSignControl |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.InactiveQuadrantAction |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.AllWindowAction |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ConcurrentCellSlopeAction |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveChartActions |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ConcurrentCellDecisions |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ConstantCellTemplates |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GridPrimitiveInputs |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.FormalLawCovariance |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.FormalLawReindex |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FormalSectorSubfamily |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.TemplateActualRelators |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ClosedLocalAssignments |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.SimultaneousActiveActions |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.CoordinateTemplateControl |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.InwardMargins |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.InitialGridActions |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.SelectedInwardActions |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ConcurrentCoordinateGates |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveStarCoherence |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.ExteriorClippedModels |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FarClippedModels |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.ExteriorPhaseGeometry |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PhaseClippedModels |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveGermActions |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrimitiveGlobalTemplates |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GlobalAlignedCentrality |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.CanonicalPolygonCovariance |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.FrameCentrality |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FrameCompatibility |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.CommonFrameTransport |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.PrivateBankFixer |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.DistinctSlotFixer |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.SmallEvenRouting |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.CanonicalSlotStars |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.TranslationTransport |
| import OAI.GroupTheory.SimpleAmenable.CentralCovers.GenerationUpstairs |
| import OAI.GroupTheory.SimpleAmenable.Amenability.LocalBarrierTransport |
| import OAI.GroupTheory.SimpleAmenable.Amenability.ChartLineTransport |
| import OAI.GroupTheory.SimpleAmenable.Configurations.PolygonVerticalStrings |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.PolygonBarFaces |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.IntervalEssSurj |
| import OAI.GroupTheory.SimpleAmenable.Homology.RestrictedPolygon |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.StageMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Homology.ProductHomologyMaps |
| import OAI.GroupTheory.SimpleAmenable.Homology.FiberSumHomology |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.TripleAllNatural |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.RegularBlockFaces |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.PointFiberMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.CellMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.TranslationFiberMonoidal |
| import OAI.GroupTheory.SimpleAmenable.Homology.RegularTotalFiniteDifferentials |
| import OAI.GroupTheory.SimpleAmenable.Configurations.SymmetricComponents |
| import OAI.GroupTheory.SimpleAmenable.Simplicial.BarBinaryBoundary |
|
|
| namespace OAI |
|
|
| section |
| open _root_.CategoryTheory _root_.OAI.CategoryTheory Limits MonoidalCategory Simplicial Opposite |
| namespace SimpleAmenable |
| open FreeChains IntervalBar.Diagram |
|
|
| attribute [local instance 1200] Rep.hV2 |
|
|
| lemma polygon_triple_homology_finite (a:ℕ) (ha:0<a) (n:ℕ) (hn:n≤5) : |
| Module.Finite ℤ ((bar₃ (C:=PolygonObject a)).homology Z n:A) := by |
| apply PolygonObject.LabelledStage.polygon_bar₃_finite_of_symmetric a ha _ n hn |
| intro q _ hq |
| exact SymmetricConfiguration.triple_homology_finite q hq |
|
|
| lemma polygon_bar_homology_finite (a:ℕ) (ha:0<a) (n:ℕ) (hn:n≤3) : |
| Module.Finite ℤ ((bar (C:=PolygonObject a)).homology Z n:A) := |
| bar_finite_of_triple (polygon_triple_homology_finite a ha) n hn |
|
|
| lemma polygon_full_low_homology_finite (a m:ℕ) (ha:0<a) (hm : 34 ≤ m) : |
| Module.Finite ℤ (groupHomology (Rep.trivial ℤ (polygonFullGroup a m) ℤ) 1) ∧ |
| Module.Finite ℤ (groupHomology (Rep.trivial ℤ (polygonFullGroup a m) ℤ) 2) := by |
| have barH1Finite := polygon_bar_homology_finite a ha 1 (by omega) |
| have barH2Finite := polygon_bar_homology_finite a ha 2 (by omega) |
| have barH3Finite := polygon_bar_homology_finite a ha 3 (by omega) |
| exact ⟨PolygonTracks.fullH1_finite_of_bar a m (by omega), |
| PolygonTracks.fullH2_finite_of_bar a m hm⟩ |
|
|
| lemma polygon_alternating_H2_finite (a m:ℕ) (ha:0<a) (hm : 205 < m) : |
| Module.Finite ℤ (groupHomology.H2 (IntegralHomology.coefficients (polygonAlternatingGroup a m))) := by |
| obtain ⟨fullH1Finite,fullH2Finite⟩:=polygon_full_low_homology_finite a m ha (by omega) |
| exact PolygonTracks.finite_alternating_H2_of_full a m hm |
|
|
| theorem main : ∃ (G : Type) (_ : Group G), |
| Infinite G ∧ Group.IsFinitelyPresented G ∧ IsSimpleGroup G ∧ FolnerAmenable G := by |
| obtain ⟨a,ha,h⟩:=fixed_track_polygon_progress |
| obtain ⟨hinfinite,hsimple,hfolner,H,hH,q,hpres,hsur,hcentral⟩:=h 206 (by omega) |
| let:=hH |
| have:=hpres |
| have:=polygonAlternatingGroup_perfect a 206 (by omega) |
| have:=polygon_alternating_H2_finite a 206 ha (by omega) |
| exact ⟨polygonAlternatingGroup a 206,inferInstance,hinfinite, |
| finitelyPresented_of_central_cover_h2 q hsur hcentral,hsimple,hfolner⟩ |
| end SimpleAmenable |
|
|
| end |
|
|
| end OAI |
|
|
