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Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds

Authors: Walter D. Neumann and Anastasiia Tsvietkova
Journal: Proc. Amer. Math. Soc. 144 (2016), 887-896
MSC (2010): Primary 57M25, 57M50, 57M27
Published electronically: October 7, 2015
MathSciNet review: 3430862
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Abstract: Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number $ k$ manifold it is enough to choose $ 3k$ specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well.

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  • [1] Colin C. Adams, Waist size for cusps in hyperbolic 3-manifolds, Topology 41 (2002), no. 2, 257-270. MR 1876890 (2003e:57023),
  • [2] A. Borel, Commensurability classes and volumes of hyperbolic $ 3$-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 1, 1-33. MR 616899 (82j:22008)
  • [3] David Coulson, Oliver A. Goodman, Craig D. Hodgson, and Walter D. Neumann, Computing arithmetic invariants of 3-manifolds, Experiment. Math. 9 (2000), no. 1, 127-152. MR 1758805 (2001c:57014)
  • [4] D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67-80. MR 918457 (89a:57020)
  • [5] Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia, A characterization of arithmetic subgroups of $ {\rm SL}(2,{\bf R})$ and $ {\rm SL}(2,{\bf C})$, Math. Nachr. 159 (1992), 245-270. MR 1237113 (94i:20088),
  • [6] Melissa L. Macasieb, Kathleen L. Petersen, and Ronald M. van Luijk, On character varieties of two-bridge knot groups, Proc. Lond. Math. Soc. (3) 103 (2011), no. 3, 473-507. MR 2827003 (2012j:57015),
  • [7] Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957 (2004i:57021)
  • [8] William W. Menasco, Polyhedra representation of link complements, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 305-325. MR 718149 (85e:57006),
  • [9] G. D. Mostow, Quasi-conformal mappings in $ n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 0236383 (38 #4679)
  • [10] Walter D. Neumann and Alan W. Reid, Arithmetic of hyperbolic manifolds, Topology '90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 273-310. MR 1184416 (94c:57024)
  • [11] Gopal Prasad, Strong rigidity of $ {\bf Q}$-rank $ 1$ lattices, Invent. Math. 21 (1973), 255-286. MR 0385005 (52 #5875)
  • [12] Alan W. Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990), no. 4, 349-352. MR 1058310 (91d:20056),
  • [13] Robert Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3) 24 (1972), 217-242. MR 0300267 (45 #9313)
  • [14] Morwen Thistlethwaite and Anastasiia Tsvietkova, An alternative approach to hyperbolic structures on link complements, Algebr. Geom. Topol. 14 (2014), no. 3, 1307-1337. MR 3190595,
  • [15] W. P. Thurston, The Geometry and Topology of Three-Manifolds, Electronic Version 1.1 (March 2002),
  • [16] A. Tsvietkova, Hyperbolic links complements, Ph.D. Thesis, University of Tennessee, 2012.
  • [17] Anastasiia Tsvietkova, Exact volume of hyperbolic 2-bridge links, Comm. Anal. Geom. 22 (2014), no. 5, 881-896. MR 3274953
  • [18] Christian K. Zickert, The volume and Chern-Simons invariant of a representation, Duke Math. J. 150 (2009), no. 3, 489-532. MR 2582103 (2011c:58053),

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Additional Information

Walter D. Neumann
Affiliation: Department of Mathematics, Barnard College, Columbia University, 2990 Broadway MC4429, New York, New York 10027

Anastasiia Tsvietkova
Affiliation: Department of Mathematics, University of California - Davis, One Shields Ave, Davis, California 95616

Keywords: Link complement, hyperbolic 3-manifold, invariant trace field, cusp, arithmetic invariants
Received by editor(s): October 10, 2014
Received by editor(s) in revised form: December 25, 2014
Published electronically: October 7, 2015
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society

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