The horoboundary and isometry group of Thurston's Lipschitz metric

We show that the horofunction boundary of Teichm\"uller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichm\"uller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.