Stable word length
= Stable word length
{title2=$\tau_S(g)$}
= Stable translation length in a group
{synonym}
For a <word metric> from a finite generating set, the <stable word length> is
$$
\tau_S(g)=\lim_{n\to\infty}\frac{|g^n|_S}{n}=\inf_{n\geq1}\frac{|g^n|_S}{n}.
$$
Subadditivity 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>.