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.