Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

摘要

We show that if M is a complete finite-volume, hyperbolic 3-manifold having exactly one cusp, and if dimZ(2) H-1(M; Z(2)) >= 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold with dim(Z2) H-1(M; Z(2)) >= 4, and if the image of the cup product map H-1(M; Z(2)) circle times H-1(M; Z(2)) -> H-2(M; Z(2)) h as dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes of the complement of an incompressible surface in M.