Measuring complexity of curves on surfaces

We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $\mathrm{\Sigma }$, including the minimum number of self-intersections, the minimum length of the words representing them in a geometric presentation of ${\pi }_1(\mathrm{\Sigma })$, and the minimum degree of the coverings of $\mathrm{\Sigma }$ to which they lift as embeddings.