In group theory, a word metric on a 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.
Read more about Word Metric: Definition, Example in A Free Group
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