Dehn surgery, homology and hyperbolic volume

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 not equal 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 of M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.