Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

We show that if $M$ is a complete, finite-volume, hyperbolic\break$3$-manifold having exactly one cusp, and if$\mathrm{dim}_{\mathbb{Z}_2}\,H_1(M;\mathbb{Z}_{2}) \geq 6$, then $M$has volume greater than $5.06$. We also show that if $M$ is a closed,orientable hyperbolic 3-manifold with $\mathrm{dim}_{\mathbb{Z}_2}\,H_1(M; \mathbb{Z}_2) \geq 4$, and if the image of the cup product map$H^1 (M; \mathbb{Z}_{2})\,{\otimes}\break H^1(M; \mathbb{Z}_{2}) \to H^2 (M;\mathbb{Z}_2)$ has dimension at most $1$, then $M$ has volume greaterthan $3.08$. The proofs of these geometric results involve newtopological results relating the Heegaard genus of a closed Hakenmanifold $M$ to the Euler characteristic of the kishkes of thecomplement of an incompressible surface in $M$.

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Published 1 January 2009