Word metric

In group theory, a word metric on a discrete group ${\displaystyle G}$ is a way to measure distance between any two elements of ${\displaystyle G}$. As the name suggests, the word metric is a metric on ${\displaystyle G}$, assigning to any two elements ${\displaystyle g}$, ${\displaystyle h}$ of ${\displaystyle G}$ a distance ${\displaystyle d(g,h)}$ that measures how efficiently their difference ${\displaystyle g^{-1}h}$ 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 ${\displaystyle G}$ must first be chosen before a word metric on ${\displaystyle G}$ 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.