Word metric

Remarks

$d_S$ is a metric. It is symmetric because $\abs{g^{-1}}_S = \abs{g}_S$, and the triangle inequality holds because $\abs{gh}_S \le \abs{g}_S + \abs{h}_S$. It is invariant under left multiplication: $d_S(kg, kh) = d_S(g, h)$ for all $k \in G$. It equals the path distance between $g$ and $h$ in the Cayley graph of $G$ with respect to $S$. If $S$ and $T$ are finite generating sets, there is a constant $C \ge 1$ with $d_T \le C d_S$ and $d_S \le C d_T$.

Example: let $G = \Z$ and $S = \set{2, 3}$. Then $\abs{1}_S = 2$: the number $1$ is not in $\set{\pm 2, \pm 3}$, so $\abs{1}_S \ge 2$, and $1 = 3 + (-2)$. So $d_S(0, 1) = 2$, while $d_{\set{1}}(0, 1) = 1$.

Non-example: in $\Z$, the set $S = \set{2}$ does not generate $\Z$. The number $1$ is not a sum of terms $\pm 2$, so $\abs{1}_S$ is not defined, and $d_S(0, 1)$ is not a real number.