On volumes and filling collections of multicurves

Abstract

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