Concept
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. The group of integers 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|. For a more illustrative example, the elements of the group can be thought of as vectors in the Cartesian plane with integer coefficients. The group is generated by the standard unit vectors , and their inverses , . The Cayley graph of is the so-called taxicab geometry. It can be pictured in the plane as an infinite square grid of city streets, where each horizontal and vertical line with integer coordinates is a street, and each point of lies at the intersection of a horizontal and a vertical street. Each horizontal segment between two vertices represents the generating vector or , depending on whether the segment is travelled in the forward or backward direction, and each vertical segment represents or .