Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

Dunfield, Nathan M.

doi:10.1007/s002220050321

Cited by 96 publications

(105 citation statements)

References 31 publications

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“…The first ensures the connectedness of the space of branched (topological) ideal triangulations of an arbitrary compact oriented 3-manifold, and the second is an uniqueness (rigidity) result for I -triangulations of cusped manifolds with maximal volume. Theorem 6.11 (See [13] or [19]) Let (T, b, w) be an arbitrary I -triangulation of a cusped 3-manifold M . Then, among all I -triangulations supported by (T, b), this is the only one such that the algebraic sum of the volumes of its I -tetrahedra equals Vol(M ).…”

Section: Cohomological Normalization Of Flattenings and Chargesmentioning

confidence: 99%

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

2005

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We introduce a family of matrix dilogarithms, which are automorphisms of C N ⊗ C N , N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 → 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N = 1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds W endowed with a flat principal P SL(2, C)-bundle ρ, and a fixed non empty link L if N > 1, and for (possibly "marked") cusped hyperbolic 3-manifolds M . When N = 1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N > 1, the invariants for M are new; those for triples (W, L, ρ) coincide with the quantum hyperbolic invariants defined in [3], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N = 1 and N > 1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N → ∞.

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Section: Cohomological Normalization Of Flattenings and Chargesmentioning

confidence: 99%

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

2005

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“…Then S D 2.a 5 C a 6 / and k2q C 11k D 2.a 5 C a 6 / D S . As in [7], this would imply that there is a nonintegral boundary slope r with j2q C 11 r j < 1. As there is no such r , we conclude that there exists i Ä 4 with a i > 0.…”

Section: Proof Of Proposition 22mentioning

confidence: 95%

Finite surgeries on three-tangle pretzel knots

Futer

Ishikawa

Kabaya

et al. 2009

Algebr. Geom. Topol.

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We classify Dehn surgeries on .p; q; r / pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit nontrivial finite surgeries are . 2; 3; 7/ and . 2; 3; 9/. Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p; q; r . We treat these using the Culler-Shalen norm of the SL.2; C/-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety. 57M05, 57M25, 57M50Dedicated to Professor Akio Kawauchi on the occasion of his 60th birthday.In [19] Mattman showed that if a hyperbolic .p; q; r / pretzel knot K admits a nontrivial finite Dehn surgery of slope s (ie, a Dehn surgery that results in a manifold of finite fundamental group) then either K D . 2; 3; 7/ and s D 17, 18 or 19, K D . 2; 3; 9/ and s D 22 or 23 or K D . 2; p; q/ where p and q are odd and 5 Ä p Ä q .In the current paper we complete the classification by proving:Theorem 1 Let K be a . 2; p; q/ pretzel knot with p , q odd and 5 Ä p Ä q . Then K admits no nontrivial finite surgery.Using the work of Agol [1] and Lackenby [16], candidates for finite surgery correspond to curves of length at most six in the maximal cusp of S 3 n K . If 7 Ä p Ä q , we will argue that only five slopes for the . 2; p; q/ pretzel knot have length six or less: the meridian and the four integral surgeries 2.p C q/ 1, 2.p C q/, 2.p C q/ C 1 and 2.p C q/ C 2. If p D 5 and q 11, a similar argument leaves seven candidates, the meridian and the six integral slopes between 2.5 C q/ 2 and 2.5 C q/ C 3.

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“…[Sw75]. Therefore, the ρ-volume vol(ρ, c) of an n-cycle c of P is well-defined (see [Du99] for further details).…”

Section: Deformations Of Representations In Hyperbolic Spacesmentioning

confidence: 95%

Systolic volume of hyperbolic manifolds and connected sums of manifolds

Sabourau

2007

Geom Dedicata

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Abstract. The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M ) satisfying the inequality vol(M ) ≥ σ(M ) sys(M ) n between the volume and the systole of every metric on M . First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.

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Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

Cited by 96 publications

References 31 publications

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

Classical and quantum dilogarithmic invariants of flatPSL(2,ℂ)–bundles over 3–manifolds

Finite surgeries on three-tangle pretzel knots

Systolic volume of hyperbolic manifolds and connected sums of manifolds

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