Séance Séminaire

Séminaire Gaston Darboux

The horofunction boundary of Thurston's metric on Teichmuller space

The horofunction boundary of a metric space was introduced by Gromov in the late 1970s. In this talk, I will describe the horofunction boundary of Teichmuller space with Thurston's Lipschitz metric. Here, the distance between two hyperbolic structures on a surface is defined to be the logarithm of the smallest Lipschitz constant with respect to the two structures, over all homeomorphisms on the surface that are isotopic to the identity. Thurston showed that this is indeed a metric, although a non-symmetric one. It turns out that the horofunction boundary of this metric is just the usual Thurston boundary of Teichmuller space. I will show how, by studying the action of isometries on this boundary, one can determine the isometry group of Thurston's metric.