Word metric

Summary

The word metric on a group with a generating set $S$ is a distance on the group. The distance from $g$ to $h$ is the least number of generators and their inverses whose product is $g^{-1} h$. It turns the group into a metric space, and it equals the path distance in the Cayley graph. For a finitely generated group, changing the finite generating set changes the metric only up to bounded factors. So the metric is well defined up to quasi-isometry. It is the starting point of geometric group theory.