If a closed, orientable hyperbolic 3-manifold M has volume at most 1.22 then H 1 .M I p / has dimension at most 2 for every prime p 2; 7, and H 1 .M I 2 / and H 1 .M I 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 .
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Agol et al. (2006) studied this question.
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