Hyperbolic.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Hyperbolic groups

A finitely generated group is hyperbolic if its induced metric space with respect to a chosen family of Group.Generators satisfies the Gromov hyperbolicity condition.

This is being upstreamed to Mathlib in leanprover-community/mathlib4#44339.

TODO: delete this file when this repository moves to a Mathlib version that contains it.

Main definitions

Group.Generators.IsHyperbolicWith: A group is δ-hyperbolic with respect to a generating family P if its induced metric space is δ-hyperbolic.
Group.IsHyperbolic: A group G is hyperbolic if there exists a finite generating family P and a constant δ such that G is δ-hyperbolic with respect to P.

Main results

[Group.IsHyperbolic G] : Group.FG G: Every hyperbolic group is finitely generated.
[Finite G] : Group.IsHyperbolic G: Every finite group is hyperbolic.

Implementation notes

The hyperbolicity condition applies directly to the word metric, without requiring the Cayley graph.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Group.Generators.IsHyperbolicWith A group is δ-hyperbolic with respect to a generating family P if its induced metric space is δ-hyperbolic line 57 source
Source documentation
A group is δ-hyperbolic with respect to a generating family P if its induced metric space is δ-hyperbolic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, Group.Generators, Real
Proof uses: Metric.IsHyperbolicWith, Group.Generators.normedGroup
2 instances, coercions and auxiliary constants left out.
```
def Generators.IsHyperbolicWith (P : Generators G ι) (δ : ℝ) : Prop :=
…
```
class Group.IsHyperbolic A group G is hyperbolic if there exists a finite generating family P and a constant δ such that G is δ-hyperbolic with respect to P line 63 source
Source documentation
A group G is hyperbolic if there exists a finite generating family P and a constant δ such that G is δ-hyperbolic with respect to P.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
class IsHyperbolic (G : Type*) [Group G] : Prop where
…
```
instance Group.(instance at line 67) line 67 source
Source documentation
Every hyperbolic group is finitely generated.
```
instance [IsHyperbolic G] : FG G := by
…
```
Mentions: Group.IsHyperbolic
instance Group.(instance at line 72) line 72 source
Source documentation
Every finite group is hyperbolic.
```
instance [Finite G] : IsHyperbolic G :=
…
```
Mentions: Group.IsHyperbolic