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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747

Dehn surgery, homology and hyperbolic volume

Ian Agol, Marc Culler and Peter B Shalen

Algebraic & Geometric Topology 6 (2006) 2297–2312

DOI: 10.2140/agt.2006.6.2297

arXiv: math.GT/0508208

Abstract

If a closed, orientable hyperbolic 3–manifold M has volume at most 1.22 then H1(M; Zp) has dimension at most 2 for every prime p not 2 or 7, and H1(M; Z2) and H1(M; Z7) 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

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