Groups as geometric objects

[bottine]

From Cornulier & de la Harpe:

Proposition 1.A.1. Let Γ be a group.

• (ct) Γ is countable if and only if it has a left-invariant metric with finite balls.
  Moreover, if d₁ , d₂ are two such metrics, the identity map (Γ, d₁ ) → (Γ, d₂ ) is a metric coarse equivalence.

Assume from now on that Γ is countable.

• (fg) Γ is finitely generated if and only if, for one (equivalently for every) metric d as in (ct), the metric space (Γ, d) is coarsely connected.
  Moreover, a finitely generated group has a left-invariant large-scale geodesic metric with finite balls (e.g. a word metric); if d₁, d₂ are two such metrics, the identity map (Γ, d₁) →(Γ, d₂) is a quasi-isometry.
• (fp) Γ is finitely presented if and only if, for one (equivalently for every) metric d as in (ct), the metric space (Γ, d) is coarsely simply connected.

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Easy consequences of these are (quoting Cornulier & de la Harpe still):

The characterizations of Proposition 1.A.1 provide conceptual proofs of some basic and well-known facts. Consider for example a countable group Γ, a subgroup of finite index ∆, a finite normal subgroup N C Γ, and a left-invariant metric d on Γ, with finite balls. Coarse connectedness and coarse simple connectedness are properties invariant by metric coarse equivalence. A straightforward verification shows that the inclusion ∆ ⊂ Γ is a metric coarse equivalence; it follows that ∆ is finitely generated (or finitely presented) if and only if Γ has the same property. It is desirable to have a similar argument for Γ and Γ/N ; for this, it is better to rephrase the characterizations (ct), (fg), and (fp) in terms of pseudo-metrics rather than in terms of metrics. “Pseudo” means that the pseudo-metric evaluated on two distinct points can be 0.

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TODO: formalize all this!