On volumes and filling collections of multicurves

Let (Formula presented.) be a surface of negative Euler characteristic and consider a finite filling collection (Formula presented.) of closed curves on (Formula presented.) in minimal position. An observation of Foulon and Hasselblatt shows that (Formula presented.) is a finite-volume hyperbolic 3-manifold, where (Formula presented.) is the projectivized tangent bundle and (Formula presented.) is the set of tangent lines to (Formula presented.). In particular, (Formula presented.) is a mapping class group invariant of the collection (Formula presented.). When (Formula presented.) 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 (Formula presented.) in a Seifert-fibered space (Formula presented.) over (Formula presented.). For such links, the volume of (Formula presented.) is coarsely comparable to expressions involving distances in the pants graph.