Cusp geometry of fibered 3-manifolds

Let $F$ be a surface and suppose that $\varphi\colon F \to F$ is a pseudo-Anosov homeomorphism, fixing a puncture $p$ of $F$. The mapping torus $M = M_\varphi$ is hyperbolic and contains a maximal cusp $C$ about the puncture $p$. We show that the area (and height) of the cusp torus $\partial C$ is equal to the stable translation distance of $\varphi$ acting on the arc complex $\scr{A}(F,p)$, up to an explicitly bounded multiplicative error. Our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces. In particular, the proof of this theorem does not use any deep results from Teichm\"uller theory, Kleinian group theory, or the coarse geometry of $\scr{A}(F,p)$. A similar result holds for quasi-Fuchsian manifolds $N\cong F\times\Bbb{R}$. In that setting, we find a combinatorial estimate for the area (and height) of the cusp annulus in the convex core of $N$, up to explicitly bounded multiplicative and additive error. As an application, we show that covers of punctured surfaces induce quasi-isometric embeddings of arc complexes.