Word metric

Definition
Suppose $$G$$ is a group and $$S$$ is a generating set for $$G$$. The word metric for $$G$$ associated with the generating set $$S$$ is defined in the following equivalent ways:


 * 1) It is a metric on $$G$$ (i.e., it makes $$G$$ into a metric space) defined as follows: the distance between $$g$$ and $$h$$ is the minimum possible length of a word using elements from $$S \cup S^{-1}$$ that evaluates to $$g^{-1}h$$.
 * 2) It is the metric on $$G$$ induced from the Cayley graph of $$G$$ with each edge of the graph having length one. More explicitly, the Cayley graph on $$G$$ is a one-dimensional simplicial complex whose geometric realization has $$G$$ as a discrete subset of it. Equip the geometric realization with a metric space structure by making each edge of length one with a suitable parametrization. Then, the induced metric on the subset $$G$$ is the word metric.