The horofunction compactification of Teichmüller spaces of surfaces with boundary

The arc metric is an asymmetric metric on the Teichmüller space T(S)T(S) of a surface S   with nonempty boundary. It is the analogue of Thurston's metric on the Teichmüller space of a surface without boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S)T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two compactifications. This generalizes a result of Walsh [20] that concerns Thurston's metric.