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.

[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.

[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.

[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.

[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.

[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.

[7]

B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.

[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.

[9]

C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.

[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.

[11]

D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., (). 

[12]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., (). 

[13]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., (). 

[14]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., (). 

[15]

D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., (). 

[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.

[17]

G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991.

[18]

G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.

[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.

[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.

[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.

[22]

R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.

[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.

[24]

Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.

[25]

S. P. Tan, Private communication, May, 2011.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.

[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.

[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.

[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.

[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.

[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.

[7]

B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.

[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.

[9]

C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.

[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.

[11]

D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., (). 

[12]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., (). 

[13]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., (). 

[14]

D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., (). 

[15]

D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., (). 

[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.

[17]

G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991.

[18]

G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.

[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.

[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.

[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.

[22]

R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.

[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.

[24]

Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.

[25]

S. P. Tan, Private communication, May, 2011.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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