Suppose S is a surface which admits a hyperbolic metric - by this I mean a complete Riemannian metric of constant negative curvature, with totally geodesic (possibly empty) boundary.
Fix some finite presentation of the fundamental group $$\pi_1(S) = \langle X\ |\ R \rangle.$$
For an element $g\in \pi_1(S)$, define the word length $|w|_X$, with respect to the generating set X, to be the minimum number of generators needed to express $w$ as a word in X.
Fact: Every homotopy class of curves in $S$ contains a unique geodesic.
Question: What's the relationship between the word length of $w$, and the hyperbolic length of the corresponding geodesic?