= Word metric
{title2=$d_S$}
{wiki}
The <word metric> measures how many allowed moves are needed to change one <group element> into another. For a finite symmetric <generating set of a group> $S$ with $1\notin S$, it is the <graph distance> between <vertices> of the <Cayley graph>, equivalently
$$
d_S(g,h)=|g^{-1}h|_S.
$$
Here <word length> counts letters in a shortest representative of $g^{-1}h$. The <path metric> on the entire <Cayley graph> also permits points inside <edges>; the <word metric> is its restriction to the <group> itself.https://people.maths.ox.ac.uk/bridson/papers/BATH97.pdf[Bridson, Section 1.6 and the remark after Example 1.8]{ref}
The <word metric> is left invariant:
$$
d_S(ag,ah)=|(ag)^{-1}(ah)|_S=|g^{-1}h|_S=d_S(g,h).
$$
This corresponds to the label-preserving <group action> on the <Cayley graph>. Symmetry follows by reversing a shortest <graph path>; concatenating routes gives the <triangle inequality>.https://webusers.imj-prg.fr/~kathryn.mann/classes/GGRminicourse.pdf[Mann, Groups, geometry, and rigidity, Definition 1.1]{ref}
Right invariance can fail in a <non-abelian group>, because right multiplication replaces $g^{-1}h$ by $a^{-1}g^{-1}ha$ through <conjugation>. In the <free group> $F(x,y)$ with $S=\{x,x^{-1},y,y^{-1}\}$, we have $d_S(1,y)=1$, whereas $d_S(x,yx)=|x^{-1}yx|_S=3$: the latter <freely reduced word> has three letters. In an <abelian group>, <conjugation> fixes each element, so right invariance does hold. <Cayley graphs of the integers> and <word metric balls in the integer lattice> give examples where the <word metric> can be calculated explicitly.
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