Hyperbolicity of link complements in Seifert-fibered spaces

Abstract

Let s γ be a link in a Seifert-fibered space M over a hyperbolic 2-orbifold O that projects injectively to a filling multi-curve of closed geodesics γ in O. We prove that the complement M s γ of s γ in M admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.

On volumes and filling collections of multicurves

“…The best‐known lower bound appears in [16, 36, 37], where the bound is given in terms of the number of homotopy classes of arcs of $$\Gamma$$ after cutting $$S$$ open along any multicurve $$\mathfrak {m}$$ and taking the maximum over such $$\mathfrak {m}$$. While this lower bound is shown to be sharp for some families of non‐simple closed curves on the modular surface [37], it is always at most $$6(3g + n)(3g -3 + n)$$ whenever $$\Gamma$$ is composed entirely of simple closed curves.…”

“…Finally, we show that this special case is coarsely comparable to the general setting using topological arguments. It is important to note that unlike the lower bound in [16, 36], our bounds control the volume of the thick‐part of $$N_{\widehat {\Gamma }}$$.…”

“…In [36], there are examples where volume grows at least as fast as $$\ell _X$$. Further, since $$i(\Gamma , \Gamma ) \leqslant K_X \ell _X(\Gamma )^2$$ for any collection of essential closed curves by [4], the intersection number bound of [16] gives a universal upper bound of $$\ell _X^2$$ for arbitrary lifts.…”

“…Let $$\overline{\Gamma }$$ be a link in a Seifert‐fibered space $$N$$ over $$S$$. In [16], the authors demonstrate that $$N_{\overline{\Gamma }}:=N\setminus \overline{\Gamma }$$ is hyperbolic whenever $$N_{\overline{\Gamma }}$$ is acylindrical and $$\overline{\Gamma }$$ is a lift of a filling collection $$\Gamma$$ of essential closed curves in minimal position via the canonical Seifert projection $$\pi :N\rightarrow S$$. Note that the acylindricity condition is needed only if one allows multiple copies of the same loop in $$\Gamma$$.…”

End-periodic homeomorphisms and volumes of mapping tori

“…A direct proof of such a comparison was later given by Brock-Bromberg [BB16] and Kojima-McShane [KM18]. Pants distance has also been related to volumes of hyperbolic 3-manifolds in another setting by Cremaschi, Rodríguez-Migueles and Yarmola [CRMY19], see also [RM20,CRM20]. In forthcoming work, Landry, Minsky, and Taylor study stretch factors of end-periodic homeomorphisms arising from depth one foliations [LMT21].…”