Left-invariant group metric ( $d (a x, a y) = d (x, y)$ )

This is a distance function satisfying the displayed identity, rather than a Riemannian metric tensor. When it induces the group's given topology, it is a compatible left-invariant group metric. Left translations are isometries.

Table of contents
- Bi-invariant group metric Left-invariant group metric
  - Affine conjugation obstruction to a bi-invariant group metric Bi-invariant group metric
- Birkhoff-Kakutani theorem Left-invariant group metric
  - Dyadic product estimate for group neighbourhoods Birkhoff-Kakutani theorem

Bi-invariant group metric ( $d (a x b, a y b) = d (x, y)$ )

A group distance is bi-invariant when both left and right translations are isometries. Conjugation then fixes the identity element and is an isometry, so each metric ball there is invariant under every inner automorphism. A compatible bi-invariant group metric therefore requires arbitrarily small conjugation-invariant neighbourhoods. This concerns a distance function, not merely a bi-invariant Riemannian metric tensor.

Affine conjugation obstruction to a bi-invariant group metric ( $D_{a} T_{b} D_{a}^{- 1} = T_{ab}$ )

In the orientation-preserving affine group of the real line, translations

T_{1/ n}

converge to the identity element, but conjugation by dilation

D_{n}

sends each one to

T_{1}

. A compatible bi-invariant group metric would give

d (e, T_{1/ n}) = d (e, T_{1}) > 0

for every

n

, contradicting convergence. The group is nonetheless metrizable in its ordinary two real coordinates.

Birkhoff-Kakutani theorem ( $G metrizable ⟺ G first countable$ )

A Hausdorff topological group admits a compatible left-invariant group metric exactly when it has a countable neighbourhood basis at the identity element. Choose symmetric neighbourhoods of the identity element

U_{n}

with

U_{n + 1}^{3} \subset U_{n}

and trivial intersection. Assign a step in

U_{n}

cost

2^{- n}

, and take the infimum of total costs of chains joining two points. The dyadic product estimate for group neighbourhoods makes this a genuine compatible metric. The Hausdorff assumption separates distinct points; without it the same construction can give only a pseudometric.

Dyadic product estimate for group neighbourhoods ( $\sum_{j} 2^{- n_{j}} \leq 2^{- n} ⟹ \prod_{j} g_{j} \in U_{n - 1}$ )

Let

U_{0} = G

, and let the

U_{n}

be decreasing symmetric sets containing the identity element with

U_{n}^{3} \subset U_{n - 1}

. If

g_{j} \in U_{n_{j}}

and their dyadic costs sum to at most

2^{- n}

for

n \geq 1

, split the ordered word at an edge crossing its half-cost point. The words on either side each have cost at most

2^{- n - 1}

. Induction on word length puts both side words in

U_{n}

, while the middle edge also belongs to

U_{n}

. Their threefold product lies in

U_{n - 1}

. This controls the short-chain collapse that would otherwise make a path-distance construction degenerate.

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