Stable word length (${\tau }_S(g)$)

For a word metric from a finite generating set, the stable word length isSubadditivity and the Fekete lemma give the limit. Conjugating $g^n$ changes its length by at most a constant, so stable word length is conjugacy invariant. It satisfies ${\tau }_S(g^k)=|k|{\tau }_S(g)$. Finite-order elements have zero stable word length; in a hyperbolic group, every infinite-order element has positive stable word length.