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 Σ$$\Sigma $$, including the minimum number of self-intersections, the minimum length of the words representing them in π1(Σ)$$\pi _1(\Sigma )$$, and the minimum degree of the coverings of Σ$$\Sigma $$ to which they lift as embeddings.