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In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitzdistance on the Teichmueller space of a (closed or punctured) surface. In thispaper we extend his work to the Teichmueller space of a surface with boundaryendowed the arc distance. In this new setting we construct a large family ofgeodesics, which generalize Thurston's stretch lines. We prove that theTeichmueller space of a surface with boundary, endowed with the arc distance,is a geodesic metric space. Furthermore, the arc distance is induced by aFinsler metric. As a corollary, we describe a new class of geodesics in theTeichmueller space of a closed/punctured surface that are not stretch lines inthe sense of Thurston.

Generalized stretch lines for surfaces with boundary