# American Institute of Mathematical Sciences

2012, 19: 97-111. doi: 10.3934/era.2012.19.97

## Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links

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

Received  June 2012 Published  November 2012

Following Riley's work, for each $2$-bridge link $K(r)$ of slope $r∈\mathbb{R}$ and an integer or a half-integer $n$ greater than $1$, we introduce the Heckoid orbifold $S(r;n)$and the Heckoid group $G(r;n)=\pi_1(S(r;n))$ of index $n$ for $K(r)$. When $n$ is an integer, $S(r;n)$ is called an even Heckoid orbifold; in this case, the underlying space is the exterior of $K(r)$, and the singular set is the lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a $4$-punctured sphere $S$ in $S(r;n)$ determined by the $2$-bridge sphere of $K(r)$, when is it null-homotopic in $S(r;n)$? (2) For two distinct essential simple loops on $S$, when are they homotopic in $S(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from $2$-bridge link groups onto Heckoid groups.
Citation: Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links. Electronic Research Announcements, 2012, 19: 97-111. doi: 10.3934/era.2012.19.97
##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] I. Agol, The classification of non-free $2$-parabolic generator Kleinian groups, Slides of talks given at Austin AMS Meeting and Budapest Bolyai conference, July 2002, Budapest, Hungary. Google Scholar [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), 21-40.  Google Scholar [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 3-manifolds'', London Math. Soc., Lecture Note Series, 329 (2006), 151-185.  Google Scholar [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.  Google Scholar [6] M. Boileau and J. Porti, Geometrization of 3-orbifolds of cyclic type, Appendix A by Michael Heusener and Porti, Astérisque No. 272 (2001).  Google Scholar [7] M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link, Math. Z., 200 (1989), 187-208.  Google Scholar [8] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78.  Google Scholar [9] B. H. Bowditch, Markoff triples and quasifuchsian groups, Proc. London Math. Soc., 77 (1998), 697-736.  Google Scholar [10] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334.  Google Scholar [11] D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, Tokyo, 2000.  Google Scholar [12] C. Gordon, Problems, Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [13] E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664-699.  Google Scholar [14] K. N. Jones and A. W. Reid, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann., 310 (1998), 235-250.  Google Scholar [15] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar [16] L. Keen and C. Series, The Riley slice of Schottky space, Proc. London Math. Soc., 69 (1994), 72-90.  Google Scholar [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), 303-316.  Google Scholar [18] D. Lee and M. Sakuma, Simple loops on $2$-bridge spheres in $2$-bridge link complements, Electron. Res. Announc. Math. Sci., 18 (2011), 97-111.  Google Scholar [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), 359-386. doi: 10.1112/plms/pdr036.  Google Scholar [20] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (I),, , ().   Google Scholar [21] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (II),, , ().   Google Scholar [22] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (III),, , ().   Google Scholar [23] D. Lee and M. Sakuma, A variation of McShane's identity for $2$-bridge links,, , ().   Google Scholar [24] D. Lee and M. Sakuma, Epimorphisms from $2$-bridge link groups onto Heckoid groups (I),, Hiroshima Math. J., ().   Google Scholar [25] D. Lee and M. Sakuma, Epimorphisms from $2$-bridge link groups onto Heckoid groups (II),, Hiroshima Math. J., ().   Google Scholar [26] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in even Heckoid orbifold for $2$-bridge links,, preliminary notes., ().   Google Scholar [27] D. Lee and M. Sakuma, A variation of McShane's identity for even Heckoid orbifolds for $2$-bridge links,, in preparation., ().   Google Scholar [28] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Springer-Verlag, Berlin, 1977.  Google Scholar [29] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph. D. Thesis, University of Warwick, 1991. Google Scholar [30] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632.  Google Scholar [31] M. Mecchia and B. Zimmermann, On a class of hyperbolic 3-orbifolds of small volume and small Heegaard genus associated to $2$-bridge links, Rend. Circ. Mat. Palermo, 49 (2000), 41-60.  Google Scholar [32] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222.  Google Scholar [33] B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc., 74 (1968), 568-571.  Google Scholar [34] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$-bridge link groups, Geom. Topol. Monogr., 14 (2008), 417-450.  Google Scholar [35] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc., 24 (1972), 217-242.  Google Scholar [36] R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc., 334 (1992), 389-409. doi: 10.1090/S0002-9947-1992-1107029-9.  Google Scholar [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., ().   Google Scholar [38] S. P. Tan, Y. L. Wong and Y. Zhang, $\SL(2,\mathbbC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110.  Google Scholar [39] 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 [40] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 119-217.  Google Scholar [41] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813.  Google Scholar [42] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\mathbbC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [43] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 37 (2008), 183-200.  Google Scholar

show all references

##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] I. Agol, The classification of non-free $2$-parabolic generator Kleinian groups, Slides of talks given at Austin AMS Meeting and Budapest Bolyai conference, July 2002, Budapest, Hungary. Google Scholar [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), 21-40.  Google Scholar [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 3-manifolds'', London Math. Soc., Lecture Note Series, 329 (2006), 151-185.  Google Scholar [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.  Google Scholar [6] M. Boileau and J. Porti, Geometrization of 3-orbifolds of cyclic type, Appendix A by Michael Heusener and Porti, Astérisque No. 272 (2001).  Google Scholar [7] M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link, Math. Z., 200 (1989), 187-208.  Google Scholar [8] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78.  Google Scholar [9] B. H. Bowditch, Markoff triples and quasifuchsian groups, Proc. London Math. Soc., 77 (1998), 697-736.  Google Scholar [10] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334.  Google Scholar [11] D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, Tokyo, 2000.  Google Scholar [12] C. Gordon, Problems, Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [13] E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664-699.  Google Scholar [14] K. N. Jones and A. W. Reid, Minimal index torsion-free subgroups of Kleinian groups, Math. Ann., 310 (1998), 235-250.  Google Scholar [15] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, 183, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar [16] L. Keen and C. Series, The Riley slice of Schottky space, Proc. London Math. Soc., 69 (1994), 72-90.  Google Scholar [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), 303-316.  Google Scholar [18] D. Lee and M. Sakuma, Simple loops on $2$-bridge spheres in $2$-bridge link complements, Electron. Res. Announc. Math. Sci., 18 (2011), 97-111.  Google Scholar [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), 359-386. doi: 10.1112/plms/pdr036.  Google Scholar [20] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (I),, , ().   Google Scholar [21] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (II),, , ().   Google Scholar [22] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in $2$-bridge link complements (III),, , ().   Google Scholar [23] D. Lee and M. Sakuma, A variation of McShane's identity for $2$-bridge links,, , ().   Google Scholar [24] D. Lee and M. Sakuma, Epimorphisms from $2$-bridge link groups onto Heckoid groups (I),, Hiroshima Math. J., ().   Google Scholar [25] D. Lee and M. Sakuma, Epimorphisms from $2$-bridge link groups onto Heckoid groups (II),, Hiroshima Math. J., ().   Google Scholar [26] D. Lee and M. Sakuma, Homotopically equivalent simple loops on $2$-bridge spheres in even Heckoid orbifold for $2$-bridge links,, preliminary notes., ().   Google Scholar [27] D. Lee and M. Sakuma, A variation of McShane's identity for even Heckoid orbifolds for $2$-bridge links,, in preparation., ().   Google Scholar [28] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Springer-Verlag, Berlin, 1977.  Google Scholar [29] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph. D. Thesis, University of Warwick, 1991. Google Scholar [30] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632.  Google Scholar [31] M. Mecchia and B. Zimmermann, On a class of hyperbolic 3-orbifolds of small volume and small Heegaard genus associated to $2$-bridge links, Rend. Circ. Mat. Palermo, 49 (2000), 41-60.  Google Scholar [32] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222.  Google Scholar [33] B. B. Newman, Some results on one-relator groups, Bull. Amer. Math. Soc., 74 (1968), 568-571.  Google Scholar [34] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$-bridge link groups, Geom. Topol. Monogr., 14 (2008), 417-450.  Google Scholar [35] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc., 24 (1972), 217-242.  Google Scholar [36] R. Riley, Algebra for Heckoid groups, Trans. Amer. Math. Soc., 334 (1992), 389-409. doi: 10.1090/S0002-9947-1992-1107029-9.  Google Scholar [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., ().   Google Scholar [38] S. P. Tan, Y. L. Wong and Y. Zhang, $\SL(2,\mathbbC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110.  Google Scholar [39] 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 [40] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 119-217.  Google Scholar [41] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813.  Google Scholar [42] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\mathbbC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [43] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 37 (2008), 183-200.  Google Scholar
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