Dehn surgery, homology and hyperbolic volume

Abstract

If a closed, orientable hyperbolic $3$–manifold $M$ has volume at most 1.22 then $H_1\left(M;{\mathbb{Z}}_p\right)$ has dimension at most $2$ for every prime $p\ne 2,7$, and $H_1\left(M;{\mathbb{Z}}_2\right)$ and $H_1\left(M;{\mathbb{Z}}_7\right)$ 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 $C$.