Hyperbolic volume and Heegaard distance

We prove (Theorem 1.5) that there exists a constant $\Lambda \gt 0$ so that if $M$ is a $(\mu, d)$-generic complete hyperbolic $3$-manifold of volume $\mathrm{Vol}(M) \lt \infty$ and $\Sigma \subset M$ is a Heegaard surface of genus $g(\Sigma) \gt \Lambda \mathrm{Vol}(M)$, then $d(\Sigma) \leq 2$, where $d(\Sigma)$ denotes the distance of $\Sigma$ as defined by Hempel. The term $(\mu, d)$-generic is described precisely in Definition 1.3; see also Remark 1.4.

The key for the proof of Theorem 1.5 is Theorem 1.8 which is of 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 $g(\Sigma) \geq 76t + 26$, then $d(\Sigma) \leq 2$.

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Published 13 May 2014