Communications in Analysis and Geometry

Volume 22 (2014)

Number 2

Hyperbolic volume and Heegaard distance

Pages: 247 – 268

DOI: http://dx.doi.org/10.4310/CAG.2014.v22.n2.a3

Authors

Tsuyoshi Kobayashi (Department of Mathematics, Nara Women’s University Kitauoya Nishimachi, Nara, Japan)

Yo’av Rieck (Department of mathematical Sciences, University of Arkansas, Fayetteville, Ark., U.S.A.)

Abstract

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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