Measuring complexity of curves on surfaces

Abstract

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

14 pages,12 figures. Exposition has been improved giving more details and examples