Marc Culler, Peter B. Shalen
Published 2007-01-24, updated 2010-10-19Version 7
If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most g. This improves the main topological result of part I, in which the the same conclusion was obtained under the stronger hypothesis that the dimension of H_1(M;Z_2) is at least 4g-1. As an application we show that if M is a closed orientable hyperbolic 3-manifold with volume at most 3.08, then H_1(M;Z_2) has dimension at most 5.
Comments: 23 pages. This version incorporates suggestions from the referee and adds a new section giving examples showing that the main theorem is almost sharp for genus 2. The examples have mod 2 homology of rank 4 and their fundamental groups contain genus 2 surface groups, but they have no closed incompressible surfaces
Singular surfaces, mod 2 homology, and hyperbolic volume, I
Treewidth, crushing, and hyperbolic volume
Hyperbolic volume, Heegaard genus and ranks of groups
![QR code for arXiv:math/0701666 [math.GT]](http://oss.arxitics.com/images/favicon.svg)