If a closed, orientable hyperbolic –manifold has volume at most 1.22 then has dimension at most for every prime , and and have dimension at most . The proof combines several deep results about hyperbolic –manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from by Dehn surgeries on .
"Dehn surgery, homology and hyperbolic volume." Algebr. Geom. Topol. 6 (5) 2297 - 2312, 2006. https://doi.org/10.2140/agt.2006.6.2297