Word Metric - Definition

Definition

Let G be a group, let S be a generating set for G, and suppose that S is closed under the inverse operation on G. A word over the set S is just a finite sequence whose entries are elements of S. The integer L is called the length of the word . Using the group operation in G, the entries of a word can be multiplied in order, remembering that the entries are elements of G. The result of this multiplication is an element in the group G which is called the evaluation of the word w. As a special case, the empty word has length zero, and its evaluation is the identity element of G.

Given an element g of G, its word norm |g| with respect to the generating set S is defined to be the shortest length of a word over S whose evaluation is equal to g. Given two elements g,h in G, the distance d(g,h) in the word metric with respect to S is defined to be . Equivalently, d(g,h) is the shortest length of a word w over S such that .

The word metric on G satisfies the axioms for a metric, and it is not hard to prove this. The proof of the symmetry axiom d(g,h) = d(h,g) for a metric uses the assumption that the generating set S is closed under inverse.