GromovPolynomialGrowth.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Gromov's theorem on groups of polynomial growth

Reference:

Wikipedia

Imports (1)

FormalConjecturesUtil

theorem GromovPolynomialGrowth.growthFunction_not_polynomial_of_infinite Documented declaration: growthFunction_not_polynomial_of_infinite line 45 source
Source documentation
Infinite groups do not satisfy polynomial growth over ℕ for any degree d because when d = 0 this reduces to the unbounded nature of growthFunction while n = 0 works when d ≠ 0. Thus a finitely-generated infinite nilpotent group would be a counter-example to Gromov's theorem when quantifying over all of ℕ, and so n = 0 should be excluded.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Infinite, Set, Set.Finite, Eq, Subgroup, Subgroup.closure, Top.top, Real, Nat, Exists, LT.lt, HMul.hMul, HPow.hPow, GromovPolynomialGrowth.GrowthFunction
Proof uses: dite, OfNat.ofNat, And, LE.le, Nat.ceil, Filter.exists_lt_of_tendsto_atTop, GromovPolynomialGrowth.tendsto_atTop_growthFunction_of_infinite, Exists.intro, lt_imp_lt_of_le_of_le, mul_le_mul_of_nonneg_right, Nat.le_ceil, pow_nonneg, Nat.cast_nonneg', le_refl, Eq.mpr, id, Eq.trans, congrFun', congrArg, pow_zero and 15 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem growthFunction_not_polynomial_of_infinite [Infinite G] {S : Set G} (hS : S.Finite)
    (h : Subgroup.closure S = ⊤) {C : ℝ} (d : ℕ) :
    ∃ (n : ℕ), C * n ^ d < GrowthFunction S n := by
…
```
Mentions: GromovPolynomialGrowth.GrowthFunction
theorem GromovPolynomialGrowth.GromovPolynomialGrowthTheorem Gromov's Polynomial Growth Theorem : A finitely generated group has polynomial growth if and only if it is virtually nilpotent sorry line 61 source
Source documentation
Gromov's Polynomial Growth Theorem : A finitely generated group has polynomial growth if and only if it is virtually nilpotent.

depends on sorry what this means
Statement uses: Group, Group.FG, Iff, GromovPolynomialGrowth.HasPolynomialGrowth, Group.IsVirtuallyNilpotent
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem GromovPolynomialGrowthTheorem [Group.FG G] :
    HasPolynomialGrowth G ↔ Group.IsVirtuallyNilpotent G := by
…
```
Mentions: GromovPolynomialGrowth.HasPolynomialGrowth