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The pentagram map in higher dimensions and KdV flows
Simple loops on 2bridge spheres in Heckoid orbifolds for 2bridge links
1.  Department of Mathematics, Pusan National University, San30 JangjeonDong, GeumjungGu, Pusan, 609735 
2.  Department of Mathematics,, Graduate School of Science, Hiroshima University, HigashiHiroshima, 7398526 
References:
[1] 
C. Adams, Hyperbolic 3manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181206. 
[2] 
I. Agol, The classification of nonfree $2$parabolic generator Kleinian groups, Slides of talks given at Austin AMS Meeting and Budapest Bolyai conference, July 2002, Budapest, Hungary. 
[3] 
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 (2004), 2140. 
[4] 
H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian groups and hyperbolic 3manifolds'', London Math. Soc., Lecture Note Series, 329 (2006), 151185. 
[5] 
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. 
[6] 
M. Boileau and J. Porti, Geometrization of 3orbifolds of cyclic type, Appendix A by Michael Heusener and Porti, Astérisque No. 272 (2001). 
[7] 
M. Boileau and B. Zimmermann, The $\pi$orbifold group of a link, Math. Z., 200 (1989), 187208. 
[8] 
B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 7378. 
[9] 
B. H. Bowditch, Markoff triples and quasifuchsian groups, Proc. London Math. Soc., 77 (1998), 697736. 
[10] 
B. H. Bowditch, A variation of McShane's identity for oncepunctured torus bundles, Topology, 36 (1997), 325334. 
[11] 
D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Threedimensional orbifolds and conemanifolds, MSJ Memoirs, 5, Mathematical Society of Japan, Tokyo, 2000. 
[12] 
C. Gordon, Problems, Workshop on Heegaard Splittings, 401411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007. 
[13] 
E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664699. 
[14] 
K. N. Jones and A. W. Reid, Minimal index torsionfree subgroups of Kleinian groups, Math. Ann., 310 (1998), 235250. 
[15] 
M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183, Birkhäuser Boston, Inc., Boston, MA, 2001. 
[16] 
L. Keen and C. Series, The Riley slice of Schottky space, Proc. London Math. Soc., 69 (1994), 7290. 
[17] 
Y. Komori and C. Series, The Riley slice revised, in "Epstein Birthday Shrift" (eds. I. Rivin, C. Rourke and C. Series), Geom. Topol. Monogr., 1 (1999), 303316. 
[18] 
D. Lee and M. Sakuma, Simple loops on $2$bridge spheres in $2$bridge link complements, Electron. Res. Announc. Math. Sci., 18 (2011), 97111. 
[19] 
D. Lee and M. Sakuma, Epimorphisms between $2$bridge link groups: Homotopically trivial simple loops on $2$bridge spheres, Proc. London Math. Soc., 104 (2012), 359386. doi: 10.1112/plms/pdr036. 
[20] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (I), arXiv:1010.2232. 
[21] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (II), arXiv:1103.0856. 
[22] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (III), arXiv:1111.3562. 
[23] 
D. Lee and M. Sakuma, A variation of McShane's identity for $2$bridge links, arXiv:1112.5859. 
[24] 
D. Lee and M. Sakuma, Epimorphisms from $2$bridge link groups onto Heckoid groups (I), Hiroshima Math. J., to appear, arXiv:1205.4631. 
[25] 
D. Lee and M. Sakuma, Epimorphisms from $2$bridge link groups onto Heckoid groups (II), Hiroshima Math. J., to appear, arXiv:1206.0429. 
[26] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in even Heckoid orbifold for $2$bridge links, preliminary notes. 
[27] 
D. Lee and M. Sakuma, A variation of McShane's identity for even Heckoid orbifolds for $2$bridge links, in preparation. 
[28] 
R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," SpringerVerlag, Berlin, 1977. 
[29] 
G. McShane, "A Remarkable Identity for Lengths of Curves," Ph. D. Thesis, University of Warwick, 1991. 
[30] 
G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607632. 
[31] 
M. Mecchia and B. Zimmermann, On a class of hyperbolic 3orbifolds of small volume and small Heegaard genus associated to $2$bridge links, Rend. Circ. Mat. Palermo, 49 (2000), 4160. 
[32] 
M. Mirzakhani, Simple geodesics and WeilPetersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179222. 
[33] 
B. B. Newman, Some results on onerelator groups, Bull. Amer. Math. Soc., 74 (1968), 568571. 
[34] 
T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$bridge link groups, Geom. Topol. Monogr., 14 (2008), 417450. 
[35] 
R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc., 24 (1972), 217242. 
[36] 
R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc., 334 (1992), 389409. doi: 10.1090/S00029947199211070299. 
[37] 
R. Riley, A personal account of the discovery of hyperbolic structures on some knot complements. With a postscript by M. B. Brin, G. A. Jones and D. Singerman, preprint. 
[38] 
S. P. Tan, Y. L. Wong and Y. Zhang, $\SL(2,\mathbbC)$ character variety of a oneholed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103110. 
[39] 
S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic conesurfaces, J. Differential Geom., 72 (2006), 73112. 
[40] 
S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 119217. 
[41] 
S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761813. 
[42] 
S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\mathbbC)$ characters of the oneholed torus, Amer. J. Math., 130 (2008), 385412. doi: 10.1353/ajm.2008.0010. 
[43] 
S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 37 (2008), 183200. 
show all references
References:
[1] 
C. Adams, Hyperbolic 3manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181206. 
[2] 
I. Agol, The classification of nonfree $2$parabolic generator Kleinian groups, Slides of talks given at Austin AMS Meeting and Budapest Bolyai conference, July 2002, Budapest, Hungary. 
[3] 
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 (2004), 2140. 
[4] 
H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian groups and hyperbolic 3manifolds'', London Math. Soc., Lecture Note Series, 329 (2006), 151185. 
[5] 
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. 
[6] 
M. Boileau and J. Porti, Geometrization of 3orbifolds of cyclic type, Appendix A by Michael Heusener and Porti, Astérisque No. 272 (2001). 
[7] 
M. Boileau and B. Zimmermann, The $\pi$orbifold group of a link, Math. Z., 200 (1989), 187208. 
[8] 
B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 7378. 
[9] 
B. H. Bowditch, Markoff triples and quasifuchsian groups, Proc. London Math. Soc., 77 (1998), 697736. 
[10] 
B. H. Bowditch, A variation of McShane's identity for oncepunctured torus bundles, Topology, 36 (1997), 325334. 
[11] 
D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Threedimensional orbifolds and conemanifolds, MSJ Memoirs, 5, Mathematical Society of Japan, Tokyo, 2000. 
[12] 
C. Gordon, Problems, Workshop on Heegaard Splittings, 401411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007. 
[13] 
E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664699. 
[14] 
K. N. Jones and A. W. Reid, Minimal index torsionfree subgroups of Kleinian groups, Math. Ann., 310 (1998), 235250. 
[15] 
M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183, Birkhäuser Boston, Inc., Boston, MA, 2001. 
[16] 
L. Keen and C. Series, The Riley slice of Schottky space, Proc. London Math. Soc., 69 (1994), 7290. 
[17] 
Y. Komori and C. Series, The Riley slice revised, in "Epstein Birthday Shrift" (eds. I. Rivin, C. Rourke and C. Series), Geom. Topol. Monogr., 1 (1999), 303316. 
[18] 
D. Lee and M. Sakuma, Simple loops on $2$bridge spheres in $2$bridge link complements, Electron. Res. Announc. Math. Sci., 18 (2011), 97111. 
[19] 
D. Lee and M. Sakuma, Epimorphisms between $2$bridge link groups: Homotopically trivial simple loops on $2$bridge spheres, Proc. London Math. Soc., 104 (2012), 359386. doi: 10.1112/plms/pdr036. 
[20] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (I), arXiv:1010.2232. 
[21] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (II), arXiv:1103.0856. 
[22] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in $2$bridge link complements (III), arXiv:1111.3562. 
[23] 
D. Lee and M. Sakuma, A variation of McShane's identity for $2$bridge links, arXiv:1112.5859. 
[24] 
D. Lee and M. Sakuma, Epimorphisms from $2$bridge link groups onto Heckoid groups (I), Hiroshima Math. J., to appear, arXiv:1205.4631. 
[25] 
D. Lee and M. Sakuma, Epimorphisms from $2$bridge link groups onto Heckoid groups (II), Hiroshima Math. J., to appear, arXiv:1206.0429. 
[26] 
D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$bridge spheres in even Heckoid orbifold for $2$bridge links, preliminary notes. 
[27] 
D. Lee and M. Sakuma, A variation of McShane's identity for even Heckoid orbifolds for $2$bridge links, in preparation. 
[28] 
R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," SpringerVerlag, Berlin, 1977. 
[29] 
G. McShane, "A Remarkable Identity for Lengths of Curves," Ph. D. Thesis, University of Warwick, 1991. 
[30] 
G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607632. 
[31] 
M. Mecchia and B. Zimmermann, On a class of hyperbolic 3orbifolds of small volume and small Heegaard genus associated to $2$bridge links, Rend. Circ. Mat. Palermo, 49 (2000), 4160. 
[32] 
M. Mirzakhani, Simple geodesics and WeilPetersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179222. 
[33] 
B. B. Newman, Some results on onerelator groups, Bull. Amer. Math. Soc., 74 (1968), 568571. 
[34] 
T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$bridge link groups, Geom. Topol. Monogr., 14 (2008), 417450. 
[35] 
R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc., 24 (1972), 217242. 
[36] 
R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc., 334 (1992), 389409. doi: 10.1090/S00029947199211070299. 
[37] 
R. Riley, A personal account of the discovery of hyperbolic structures on some knot complements. With a postscript by M. B. Brin, G. A. Jones and D. Singerman, preprint. 
[38] 
S. P. Tan, Y. L. Wong and Y. Zhang, $\SL(2,\mathbbC)$ character variety of a oneholed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103110. 
[39] 
S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic conesurfaces, J. Differential Geom., 72 (2006), 73112. 
[40] 
S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 119217. 
[41] 
S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761813. 
[42] 
S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\mathbbC)$ characters of the oneholed torus, Amer. J. Math., 130 (2008), 385412. doi: 10.1353/ajm.2008.0010. 
[43] 
S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 37 (2008), 183200. 
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