Singular surfaces, mod 2 homology, and hyperbolic volume, I
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This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. This result should be viewed as an analogue of Dehn's Lemma for π_1-injective singular surfaces. The geometric application states that if M is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric theorem combines the topological theorem with several deep geometric results, including the Marden tameness conjecture,recently established by Agol and by Calegari-Gabai; a co-volume estimate for 3-tame, 3-free Kleinian groups due to Anderson, Canary, Culler and Shalen; and a volume estimate for hyperbolic Haken manifolds recently proved by Agol, Storm and W. Thurston.
Ian Agol has been added as a co-author. The main topological result has been considerably strengthened. It now applies to singular surfaces of any genus, and in the genus 2 case considered in the earlier version the homology bound has been lowered from 11 to 7. In addition, the arguments have been substantially simplified
Ian Agol has been added as a co-author. The main topological result has been considerably strengthened. It now applies to singular surfaces of any genus, and in the genus 2 case considered in the earlier version the homology bound has been lowered from 11 to 7. In addition, the arguments have been substantially simplified