# American Institute of Mathematical Sciences

2011, 18: 97-111. doi: 10.3934/era.2011.18.97

## Simple loops on 2-bridge spheres in 2-bridge link complements

 1 Department of Mathematics, Pusan National University, San-30 Jangjeon-Dong, Geumjung-Gu, Pusan, 609-735, South Korea 2 Department of Mathematics,, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Received  April 2011 Revised  June 2011 Published  August 2011

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity.
Citation: Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in 2-bridge link complements. Electronic Research Announcements, 2011, 18: 97-111. doi: 10.3934/era.2011.18.97
##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] H. Akiyoshi, H. Miyachi and M. Sakuma, A refinement of McShane's identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III, Contemporary Math., 355, Amer. Math. Soc., Providence, RI, (2004), 21-40.  Google Scholar [3] H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian Groups," London Math. Soc. Lecture Note Series, 329, Cambridge Univ. Press, Cambridge, (2006), 151-185.  Google Scholar [4] H. Akiyoahi, M. Sakuma, M. Wada and Y. Yamashita, "Punctured Torus Groups and 2-Bridge Knot Groups (I)," Lecture Notes in Mathematics, 1909, Springer, Berlin, 2007.  Google Scholar [5] K. I. Appel and P. E. Schupp, The conjugacy problem for the group of any tame alternating knot is solvable, Proc. Amer. Math. Soc., 33 (1972), 329-336. doi: 10.1090/S0002-9939-1972-0294460-X.  Google Scholar [6] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78. doi: 10.1112/blms/28.1.73.  Google Scholar [7] B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.  Google Scholar [8] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334. doi: 10.1016/0040-9383(96)00017-1.  Google Scholar [9] C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [10] K. Johnsgard, The conjugacy problem for the groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc., 349 (1997), 857-901. doi: 10.1090/S0002-9947-97-01617-6.  Google Scholar [11] D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., ().   Google Scholar [12] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., ().   Google Scholar [13] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., ().   Google Scholar [14] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., ().   Google Scholar [15] D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., ().   Google Scholar [16] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 89, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991. Google Scholar [18] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.  Google Scholar [19] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222. doi: 10.1007/s00222-006-0013-2.  Google Scholar [20] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between 2-bridge link groups, Teh Zieschang Gedenkschrift, Geom. Topol. Monogr., 14, Geom. Topol. Publ., Coventry, (2008), 417-450.  Google Scholar [21] J.-P. Préaux, Conjugacy problems in groups of oriented geometrizable 3-manifolds, Topology, 45 (2006), 171-208. doi: 10.1016/j.top.2005.06.002.  Google Scholar [22] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.  Google Scholar [23] M. Sakuma, Variations of McShane's identity for the Riley slice and 2-bridge links, In "Hyperbolic Spaces and Related Topics" (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku, 1104 (1999), 103-108.  Google Scholar [24] Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.  Google Scholar [25] S. P. Tan, Private communication, May, 2011. Google Scholar [26] S. P. Tan, Y. L. Wong and Y. Zhang, The $\SL(2,\CC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110. doi: 10.1090/S1079-6762-05-00153-8.  Google Scholar [27] S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom., 72 (2006), 73-112.  Google Scholar [28] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 199-217. doi: 10.1007/s10711-006-9069-9.  Google Scholar [29] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813. doi: 10.1016/j.aim.2007.09.004.  Google Scholar [30] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\CC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [31] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 237 (2008), 183-200. doi: 10.2140/pjm.2008.237.183.  Google Scholar [32] C. M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc., 30 (1971), 22-26. doi: 10.1090/S0002-9939-1971-0279169-X.  Google Scholar

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##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] H. Akiyoshi, H. Miyachi and M. Sakuma, A refinement of McShane's identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III, Contemporary Math., 355, Amer. Math. Soc., Providence, RI, (2004), 21-40.  Google Scholar [3] H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian Groups," London Math. Soc. Lecture Note Series, 329, Cambridge Univ. Press, Cambridge, (2006), 151-185.  Google Scholar [4] H. Akiyoahi, M. Sakuma, M. Wada and Y. Yamashita, "Punctured Torus Groups and 2-Bridge Knot Groups (I)," Lecture Notes in Mathematics, 1909, Springer, Berlin, 2007.  Google Scholar [5] K. I. Appel and P. E. Schupp, The conjugacy problem for the group of any tame alternating knot is solvable, Proc. Amer. Math. Soc., 33 (1972), 329-336. doi: 10.1090/S0002-9939-1972-0294460-X.  Google Scholar [6] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78. doi: 10.1112/blms/28.1.73.  Google Scholar [7] B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.  Google Scholar [8] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334. doi: 10.1016/0040-9383(96)00017-1.  Google Scholar [9] C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [10] K. Johnsgard, The conjugacy problem for the groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc., 349 (1997), 857-901. doi: 10.1090/S0002-9947-97-01617-6.  Google Scholar [11] D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., ().   Google Scholar [12] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., ().   Google Scholar [13] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., ().   Google Scholar [14] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., ().   Google Scholar [15] D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., ().   Google Scholar [16] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 89, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991. Google Scholar [18] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.  Google Scholar [19] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222. doi: 10.1007/s00222-006-0013-2.  Google Scholar [20] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between 2-bridge link groups, Teh Zieschang Gedenkschrift, Geom. Topol. Monogr., 14, Geom. Topol. Publ., Coventry, (2008), 417-450.  Google Scholar [21] J.-P. Préaux, Conjugacy problems in groups of oriented geometrizable 3-manifolds, Topology, 45 (2006), 171-208. doi: 10.1016/j.top.2005.06.002.  Google Scholar [22] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.  Google Scholar [23] M. Sakuma, Variations of McShane's identity for the Riley slice and 2-bridge links, In "Hyperbolic Spaces and Related Topics" (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku, 1104 (1999), 103-108.  Google Scholar [24] Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.  Google Scholar [25] S. P. Tan, Private communication, May, 2011. Google Scholar [26] S. P. Tan, Y. L. Wong and Y. Zhang, The $\SL(2,\CC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110. doi: 10.1090/S1079-6762-05-00153-8.  Google Scholar [27] S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom., 72 (2006), 73-112.  Google Scholar [28] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 199-217. doi: 10.1007/s10711-006-9069-9.  Google Scholar [29] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813. doi: 10.1016/j.aim.2007.09.004.  Google Scholar [30] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\CC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [31] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 237 (2008), 183-200. doi: 10.2140/pjm.2008.237.183.  Google Scholar [32] C. M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc., 30 (1971), 22-26. doi: 10.1090/S0002-9939-1971-0279169-X.  Google Scholar
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