In group theory, a word metric on a discrete group G {\displaystyle G} is a way to measure distance between any two elements of G {\displaystyle G} . As the name suggests, the word metric is a metric on G {\displaystyle G} , assigning to any two elements g {\displaystyle g} , h {\displaystyle h} of G {\displaystyle G} a distance d ( g , h ) {\displaystyle d(g,h)} that measures how efficiently their difference g − 1 h {\displaystyle g^{-1}h} can be expressed as a word whose letters come from a generating set for the group. The word metric on G is very closely related to the Cayley graph of G: the word metric measures the length of the shortest path in the Cayley graph between two elements of G. A generating set for G {\displaystyle G} must first be chosen before a word metric on G {\displaystyle G} is specified. Different choices of a generating set will typically yield different word metrics. While this seems at first to be a weakness in the concept of the word metric, it can be exploited to prove theorems about geometric properties of groups, as is done in geometric group theory.
Examples
The group of integers Z {\displaystyle \mathbb {Z} }
The group of integers Z {\displaystyle \mathbb {Z} } is generated by the set {-1,+1}. The integer -3 can be expressed as -1-1-1+1-1, a word of length 5 in these generators. But the word that expresses -3 most efficiently is -1-1-1, a word of length 3. The distance between 0 and -3 in the word metric is therefore equal to 3. More generally, the distance between two integers m and n in the word metric is equal to |m-n|, because the shortest word representing the difference m-n has length equal to |m-n|.
The group Z ⊕ Z {\displaystyle \mathbb {Z} \oplus \mathbb {Z} }
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