2006
DOI: 10.2140/agt.2006.6.2297
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Dehn surgery, homology and hyperbolic volume

Abstract: If a closed, orientable hyperbolic 3--manifold M has volume at most 1.22 then H_1(M;Z_p) has dimension at most 2 for every prime p not 2 or 7, and H_1(M;Z_2) and H_1(M;Z_7) have dimension at most 3. The proof combines several deep results about hyperbolic 3--manifolds. The strategy is to compare the volume of a tube about a shortest closed geodesic C in M with the volumes of tubes about short closed geodesics in a sequence of hyperbolic manifolds obtained from M by Dehn surgeries on C.Comment: This is the vers… Show more

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Cited by 28 publications
(76 citation statements)
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“…The vertices of T are the components of X 2 and of X 3 . Two vertices of T are joined by an edge if they correspond to subsets of |K 3 | − |K 0 |, one of which is a component of X 2 and one a component of X 3 , and if some simplex contained in one of these sets is a face of a simplex contained in one of the other sets.…”
Section: Introductionmentioning
confidence: 99%
“…The vertices of T are the components of X 2 and of X 3 . Two vertices of T are joined by an edge if they correspond to subsets of |K 3 | − |K 0 |, one of which is a component of X 2 and one a component of X 3 , and if some simplex contained in one of these sets is a face of a simplex contained in one of the other sets.…”
Section: Introductionmentioning
confidence: 99%
“…The following geometric result is made possible by recent advances in Kleinian groups, and generalizes [4,Lemma 4.3]. Lemma 3.2 Let h˛;ˇ; i be a free bicuspid group, with maximal horocusp H. Then Vol.H/ .…”
Section: Background and Definitionsmentioning
confidence: 97%
“…Next, we proceed as in Lemma 4.3 of [4]. Suppose that .P j / is an infinite sequence of distinct hyperbolic manifolds obtained by Dehn filling N i along S i using Thurston's hyperbolic Dehn surgery theorem for geometrically finite manifolds (see Bonahon and Otal [8] or Comar [17]).…”
Section: Background and Definitionsmentioning
confidence: 99%
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“…By Theorem A.2, this two-generator group must be conjugate to G 3,6 from Table 1. After conjugation, we assume that G = G 3,6 .…”
Section: Appendix a Volume Bounds For Small Collar Radiusmentioning
confidence: 99%