Cusp geometry of fibered 3-manifolds

TL;DR

This paper establishes a direct, explicit relationship between the cusp geometry of fibered 3-manifolds and the stable translation distance of the monodromy on the arc complex, avoiding deep Teichmüller theory.

Contribution

It provides a new elementary proof linking cusp geometry to combinatorial data of the monodromy, with explicit bounds, applicable to both hyperbolic and quasi-Fuchsian manifolds.

Findings

Cusp area and height are proportional to stable translation distance of the monodromy.

Elementary proof avoids deep Teichmüller and Kleinian theory.

Covers of punctured surfaces induce quasi-isometric embeddings of arc complexes.

Abstract

Let F be a surface and suppose that \phi: F -> F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = M_\phi is hyperbolic and contains a maximal cusp C about the puncture p. We show that the area (and height) of the cusp torus bounding C is equal to the stable translation distance of \phi acting on the arc complex 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 in Teichmueller theory, Kleinian group theory, or the coarse geometry of A(F,p). A similar result holds for quasi-Fuchsian manifolds N = (F x 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 an explicitly bounded multiplicative and additive error. As an application, we show that covers of punctured surfaces induce quasi-isometric embeddings of arc complexes.