Communications in Analysis and Geometry

Volume 17 (2009)

Number 5

Volume and topology of bounded and closed hyperbolic $3$-manifolds

Pages: 797 – 849

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n5.a1

Authors

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

Let $N$ be a compact, orientable hyperbolic $3$-manifold with$\partial N$ a connected totally geodesic surface of genus $2$. If$N$ has Heegaard genus at least $5$, then its volume is greater than$6.89$. The proof of this result uses thefollowing dichotomy: either the shortest return path(defined by Kojima–Miyamoto) of $N$ is long, or $N$ has an embedded codimension-$0$submanifold $X$ with incompressible boundary $T \sqcup \partial N$,where $T$ is the frontier of $X$ in $N$, which is not a book of $I$-bundles.As an application of this result, we show that if $M$ is a closed,orientable hyperbolic $3$-manifold with $\mathrm{dim}_{\mathbb{Z}_2} H_1(M; \mathbb{Z}_2) \geq 5$, and if thecup product map $H^1 (M;\mathbb{Z}_2) \otimes H^1(M;\mathbb{Z}_2)\rightarrow H^2(M;\mathbb{Z}_2)$ has image of dimension at most one,then $M$ has volume greater than $3.44$.

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