The horofunction compactification of a metric space keeps track of the possible limits of balls whose centers go off to infinity. This construction was introduced by Gromov, and although it is usually hard to visualize, it has proved to be a useful tool for studying negatively curved spaces. For example, Tiozzo has recently used it to determine the Poisson boundary of random walks on weakly hyperbolic groups. In the context of Teichmüller space, the horofunction compactification with respect to the Thurston metric is isomorphic to the Thurston compactification, and with respect to the Teichmüller metric it is isomorphic to the Gardiner-Masur compactification, which gives these constructions a certain naturality. In this talk, I will explain how the horofunction compactification with respect to the Teichmüller metric can be seen as a refinement of the visual compactification. I will also list the few cases where these two compactifications are isomorphic. Time permitting, I will give further consequences of this perspective.

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