 Word length

The word length $|g{|}_S$ is the least number of letters from a symmetric generating set of a group $S$ whose product is $g$. The empty word has length zero. It obeys $|gh{|}_S\le |g{|}_S+|h{|}_S$, $|g^{-1}{|}_S=|g{|}_S$ and $d_S(g,h)=|g^{-1}h{|}_S$. The generating set may be infinite; finiteness is needed only for additional conclusions such as finite metric balls.