Hyperbolic volume and Heegaard distance

Abstract

We prove (Theorem~1.5) that there exists a constant $\Lambda >0$ so that if $M$ is a $(\mu ,d)$-generic complete hyperbolic 3-manifold of volume $\vol[M] < \infty$ and $\Sigma \subset M$ is a Heegaard surface of genus $g(Σ) > Λ\vol[M]$, then $d(\Sigma )\le 2$, where $d(\Sigma )$ denotes the distance of $\Sigma$ as defined by Hempel. The key for the proof of the main result is Theorem~1.8 which is on independent interest. There we prove that if $M$ is a compact 3-manifold that can be triangulated using at most $t$ tetrahedra (possibly with missing or truncated vertices), and $\Sigma$ is a Heegaard surface for $M$ with $g(\Sigma )\ge 76t+26$, then $d(\Sigma )\le 2$.

12pages, 3 figures

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