Word Metric - Example in A Free Group

Example in A Free Group

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.