Thin triangles and a multiplicative ergodic theorem for Teichm\"uller geometry

We show that, in the Teichm"uller 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 Teichm"uller space. In particular, an application is worked out for the action of the mapping class group: we show that geodesics track random walks sublinearly.