Geodesic
Dehn surgery, homology and hyperbolic volume
Agol, Ian1 ; Culler, Marc1 ; Shalen, Peter B1
1Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S Morgan St, Chicago, IL 60607-7045, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 5, pp. 2297-2312
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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≠2,7, 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.

DOI : 10.2140/agt.2006.6.2297
Keywords: hyperbolic manifold, volume, homology, drilling, Dehn surgery

Agol, Ian  1   ; Culler, Marc  1   ; Shalen, Peter B  1

1 Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S Morgan St, Chicago, IL 60607-7045, USA 
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Agol, Ian; Culler, Marc; Shalen, Peter B. Dehn surgery, homology and hyperbolic volume. Algebraic and Geometric Topology, Tome 6 (2006) no. 5, pp. 2297-2312. doi: 10.2140/agt.2006.6.2297

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