Volume 6, issue 5 (2006)

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Dehn surgery, homology and hyperbolic volume

Ian Agol, Marc Culler and Peter B Shalen

Algebraic & Geometric Topology 6 (2006) 2297–2312

arXiv: math.GT/0508208

Abstract

If a closed, orientable hyperbolic 3–manifold M has volume at most 1.22 then H1(M; p) has dimension at most 2 for every prime p2,7, and H1(M; 2) and H1(M; 7) have dimension at most 3. The proof combines several deep results about hyperbolic 3–manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.

Keywords
hyperbolic manifold, volume, homology, drilling, Dehn surgery
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57M27
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Publication
Received: 14 July 2006
Accepted: 1 November 2006
Published: 8 December 2006
Authors
Ian Agol
Department of Mathematics, Statistics, and Computer Science (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago, IL 60607-7045
USA
Marc Culler
Department of Mathematics, Statistics, and Computer Science (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago, IL 60607-7045
USA
Peter B Shalen
Department of Mathematics, Statistics, and Computer Science (M/C 249)
University of Illinois at Chicago
851 S Morgan St
Chicago, IL 60607-7045
USA