Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

Details Incompressible surfaces, hyperbolic volume, Heegaard genus and homology Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

2008-07-29

arxiv.org/abs/0807.4718v2

Mathematics

Geometric Topology

24 pages, some typographical errors have been corrected and a few passages were reworded to improve clarity

Scientific paper

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has 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 (i.e guts) of the complement of an incompressible surface in M.

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