Dehn surgery, homology and hyperbolic volume

Agol, Ian; Culler, Marc; Shalen, Peter B

doi:10.2140/agt.2006.6.2297

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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

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 $H_{1} (M; ℤ_{p})$ has dimension at most $2$ for every prime $p \neq 2, 7$ , and $H_{1} (M; ℤ_{2})$ and $H_{1} (M; ℤ_{7})$ 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 $C \subset M$ 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

Volume 6, issue 5 (2006)