page_title: Word metric
section_title: Example in a free group
hierarchical_section_title: Word metric / Example in a free group
context_page_description: In group theory, a word metric on a discrete group is a way to measure distance between any two elements of. As the name suggests, the word metric is a metric on, assigning to any two elements, of a distance that measures how efficiently their difference 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 must first be chosen before a word metric on 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.
context_section_description: Suppose that F is the free group on the two element set . A word w in the symmetric generating set  is said to be reduced if the letters  do not occur next to each other in w, nor do the letters . Every element  is represented by a unique reduced word, and this reduced word is the shortest word representing g. For example, since the word  is reduced and has length 2, the word norm of  equals 2, so the distance in the word norm between  and  equals 2. This can be visualized in terms of the Cayley graph, where the shortest path between b and a has length 2.
caption: In the free group on the two element set {a,b}, the distance between a and b in the word metric equals 2