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Volume change under drilling. (English) Zbl 1031.57014

Given a complete hyperbolic 3-manifold M containing an embedded tubular \(R\)-neighborhood of closed geodesic \(\gamma\), the author estimates the volume of a complete hyperbolic metric on the complement \(M_{\gamma}=M\backslash\gamma\): \[ \text{Vol}(M_{\gamma})\leq (\coth R)^{1/2}(\coth 2R)^{1/2}\text{Vol}(M) . \] As a corollary he shows that the smallest volume orientable hyperbolic 3-manifold has volume \(>0.32\).

MSC:

57M50 General geometric structures on low-dimensional manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C22 Geodesics in global differential geometry

Software:

SnapPea
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References:

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