On volumes and filling collections of multicurves

TL;DR

This paper explores the relationship between volumes of hyperbolic 3-manifolds derived from filling collections of curves on surfaces and distances in Teichmüller space, introducing new links between geometric invariants.

Contribution

It establishes coarse comparisons between hyperbolic volumes and Weil-Petersson distances for filling curve collections, using stratified hyperbolic links in Seifert-fibered spaces.

Findings

Volume of the hyperbolic 3-manifold is comparable to Weil-Petersson distance.

Volumes of stratified hyperbolic links relate to distances in the pants graph.

The approach connects hyperbolic geometry with Teichmüller theory.

Abstract

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $\Gamma$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hat{\Gamma}$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hat\Gamma$ is the set of tangent lines to $\Gamma$. In particular, $vol(PT(S) \setminus \hat{\Gamma})$ is a mapping class group invariant of the collection $\Gamma$. When $\Gamma$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichm\"uller space. Our main tool is the study of stratified hyperbolic links $\bar\Gamma$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\bar\Gamma$ is coarsely comparable to expressions involving distances in the pants graph.