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.
We construct a locally hyperbolic 3-manifold
M
M
such that
π
1
(
M
)
\pi _1(M)
has no divisible subgroups. We then show that
M
M
is not homotopy equivalent to any complete hyperbolic manifold.
We construct a locally hyperbolic 3-manifold M such that π 1 (M ) has no divisible subgroups. We then show that M is not homotopy equivalent to any complete hyperbolic manifold.Notation: We use for homotopic and by π 0 (X) we intend the connected components of X. With Σ g,k we denote the genus g orientable surface with k boundary components. By N → M we denote embeddings while S M denotes immersions.
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 PT(S)∖trueΓ̂$PT(S) \setminus \widehat {\Gamma }$ is a finite‐volume hyperbolic 3‐manifold, where PTfalse(Sfalse)$PT(S)$ is the projectivized tangent bundle and normalΓ̂$\widehat \Gamma$ is the set of tangent lines to Γ$\Gamma$. In particular, volfalse(PT(S)∖trueΓ̂false)$vol(PT(S) \setminus \widehat {\Gamma })$ 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 normalΓ¯$\overline{\Gamma }$ 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.
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