Title: Best Lipschitz maps and geodesic laminations Abstract. In a 1995 preprint, William Thurston outlined a Teichmuller theory for hyperbolic surfaces based on maps between surfaces which minimize the Lipschitz constant. In this talk I will describe an attempt (joint with Karen Uhlenbeck) to provide an analytic foundation of his theory. More precisely, I will describe a notion of infinity harmonic map whose dual geodesic lamination contains (is conjecturally equal) to Thurston's canonical lamination. I will also describe some transverse measures with values in the Lie algebra that naturally arise as conservation laws from the symmetries of the domain and the target and which to my knowledge have not been considered before by the Thurston school. All is work in progress.