= Bi-invariant group metric
{title2=$d(axb,ayb)=d(x,y)$}
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.
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