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# 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

#### 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 $3.44$.