A variation of McShane’s identity for 2–bridge links

Lee, Donghi¹ ; Sakuma, Makoto²

¹ Department of Mathematics, Pusan National University, Pusan 609-735, South Korea
² Department of Mathematics, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Geometry & topology, Tome 17 (2013) no. 4, pp. 2061-2101.

Résumé

We give a variation of McShane’s identity, which describes the cusp shape of a hyperbolic $2$–bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL\left(2,ℂ\right)$–characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of $2$–bridge link complements.

Détail

DOI : 10.2140/gt.2013.17.2061

Classification : 20F06, 57M25, 57M50
Keywords: 2-bridge link, 2-bridge knot, punctured torus, end invariant, McShane's identity

Bibliographie

[1] W Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics 820, Springer (1980)

[2] H Akiyoshi, H Miyachi, M Sakuma, A refinement of McShane's identity for quasifuchsian punctured torus groups, from: "In the tradition of Ahlfors and Bers, III" (editors W Abikoff, A Haas), Contemp. Math. 355, Amer. Math. Soc. (2004) 21

[3] H Akiyoshi, H Miyachi, M Sakuma, Variations of McShane's identity for punctured surface groups, from: "Spaces of Kleinian groups" (editors Y N Minsky, M Sakuma, C Series), London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press (2006) 151

[4] H Akiyoshi, M Sakuma, M Wada, Y Yamashita, Punctured torus groups and 2–bridge knot groups. I, Lecture Notes in Mathematics 1909, Springer (2007)

[5] B H Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc. 28 (1996) 73

[6] B H Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology 36 (1997) 325

[7] B H Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. 77 (1998) 697

[8] W Dicks, M Sakuma, On hyperbolic once-punctured-torus bundles. III. Comparing two tessellations of the complex plane, Topology Appl. 157 (2010) 1873

[9] D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67

[10] F Guéritaud, Géométrie hyperbolique effective et triangulations idéales canoniques en dimension trois, PhD thesis, Université Paris Sud – Paris XI (2006)

[11] F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239

[12] D Lee, M Sakuma, Simple loops on 2–bridge spheres in 2–bridge link complements, Electron. Res. Announc. Math. Sci. 18 (2011) 97

[13] D Lee, M Sakuma, Epimorphisms between 2–bridge link groups: homotopically trivial simple loops on 2–bridge spheres, Proc. Lond. Math. Soc. 104 (2012) 359

[14] D Lee, M Sakuma, Homotopically equivalent simple loops on 2–bridge spheres in 2–bridge link complements (I), to appear in Geom. Dedicata

[15] D Lee, M Sakuma, Homotopically equivalent simple loops on 2–bridge spheres in 2–bridge link complements (II), to appear in Geom. Dedicata

[16] D Lee, M Sakuma, Homotopically equivalent simple loops on 2–bridge spheres in 2–bridge link complements (III), to appear in Geom. Dedicata

[17] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford University Press (1998)

[18] G Mcshane, A remarkable identity for lengths of curves, PhD thesis, University of Warwick (1991)

[19] G Mcshane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998) 607

[20] M Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007) 179

[21] T Nakanishi, A series associated to generating pairs of a once punctured torus group and a proof of McShane's identity, Hiroshima Math. J. 41 (2011) 11

[22] T Ohtsuki, R Riley, M Sakuma, Epimorphisms between 2–bridge link groups, from: "The Zieschang Gedenkschrift" (editors M Boileau, M Scharlemann, R Weidmann), Geom. Topol. Monogr. 14 (2008) 417

[23] M Sakuma, Variations of McShane's identity for the Riley slice and 2–bridge links, Sūrikaisekikenkyūsho Kōkyūroku 1104 (1999) 103

[24] M Sakuma, Epimorphisms between 2–bridge knot groups from the view point of Markoff maps, from: "Intelligence of low dimensional topology 2006" (editors J S Carter, S Kamada, L H Kauffman, A Kawauchi, T Kohno), Ser. Knots Everything 40, World Sci. Publ., Hackensack, NJ (2007) 279

[25] M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. 21 (1995) 393

[26] H Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956) 133

[27] M Sheingorn, Characterization of simple closed geodesics on Fricke surfaces, Duke Math. J. 52 (1985) 535

[28] S P Tan, Y L Wong, Y Zhang, The $\mathrm{SL}(2,\mathbb C)$ character variety of a one-holed torus, Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 103

[29] S P Tan, Y L Wong, Y Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom. 72 (2006) 73

[30] S P Tan, Y L Wong, Y Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata 119 (2006) 199

[31] S P Tan, Y L Wong, Y Zhang, End invariants for $\mathrm{SL}(2,\mathbb C)$ characters of the one-holed torus, Amer. J. Math. 130 (2008) 385

[32] S P Tan, Y L Wong, Y Zhang, Generalized Markoff maps and McShane's identity, Adv. Math. 217 (2008) 761

[33] S P Tan, Y L Wong, Y Zhang, McShane's identity for classical Schottky groups, Pacific J. Math. 237 (2008) 183

[34] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)

[35] M Wada, OPTi

[36] J Weeks, SnapPea

[37] J R Weeks, Convex hulls and isometries of cusped hyperbolic $3$–manifolds, Topology Appl. 52 (1993) 127

[38] Y Yamashita, calc.py (2012)