2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75956We show that, in the Teichmüller metric, "thin-framed triangles are thin"---that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on Teichmüller space. In particular, an application is worked out for the action of the mapping class group: we show that geodesics track random walks sublinearly.23 pagesGeometric TopologyDynamical Systems30F60; 37A30text