On length spectrum metrics and weak metrics on Teichmüller spaces of surfaces with boundary

Abstract

We define and study natural metrics and weak metrics on the Teichmller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmller space which we call 0 -relative -thick parts, for > 0 and 0 > 0. We compare the topologies defined by these metrics on Teichmller space and we study divergence to infinity with respect to these various metrics.