Communications in Analysis and Geometry

Volume 17 (2009)

Number 2

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

Pages: 155 – 184

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n2.a1

Authors

Marc Culler (Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago)

Jason Deblois (Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago)

Peter B. Shalen (Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago)

Abstract

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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