Singular surfaces, mod 2 homology, and hyperbolic volume, I

Agol, Ian; Culler, Marc; Shalen, Peter

doi:10.1090/S0002-9947-10-04362-X

Singular surfaces, mod 2 homology, and hyperbolic volume, I
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by Ian Agol, Marc Culler and Peter B. Shalen PDF

Trans. Amer. Math. Soc. 362 (2010), 3463-3498 Request permission

Abstract:

If $M$ is a simple, closed, orientable $3$-manifold such that $\pi _1(M)$ contains a genus-$g$ surface group, and if $H_1(M;\mathbb {Z}_2)$ has rank at least $4g-1$, we show that $M$ contains an embedded closed incompressible surface of genus at most $g$. As an application we show that if $M$ is a closed orientable hyperbolic $3$-manifold of volume at most $3.08$, then the rank of $H_1(M;\mathbb {Z}_2)$ is at most $6$.

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