Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative e{\epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and e{\epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.