Dehn surgery, homology and hyperbolic volume
Ian Agol, Marc Culler, Peter B Shalen
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 in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.
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