Often in the instance of negative curvature, we find rigidity by passing to the boundary. In this talk, I will explain one instance of this phenomenon: mapping class groups of hyperbolic surfaces with a marked point. Such a group can be identified with the group $\mathrm{Aut}(\pi_1(S))$ of automorphisms of the fundamental group of the surface, which acts naturally on the Gromov boundary of $\pi_1(S)$, a topological circle. In new work with M. Wolff, we show any nontrivial action of $\mathrm{Aut}(\pi_1(S))$ on the circle is semiconjugate to this natural boundary action, answering a question posed by Farb. I'll explain the context for this result, and how simple closed curves on the surface play a key role in the proof.