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.   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, (2002).   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, 355 (2004), 21.   Google Scholar

[4]

H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups,, Proceedings of the Workshop, 329 (2006), 151.   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 (2007).   Google Scholar

[6]

M. Boileau and J. Porti, Geometrization of 3-orbifolds of cyclic type,, Appendix A by Michael Heusener and Porti, (2001).   Google Scholar

[7]

M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link,, Math. Z., 200 (1989), 187.   Google Scholar

[8]

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

[9]

B. H. Bowditch, Markoff triples and quasifuchsian groups,, Proc. London Math. Soc., 77 (1998), 697.   Google Scholar

[10]

B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles,, Topology, 36 (1997), 325.   Google Scholar

[11]

D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds,, MSJ Memoirs, 5 (2000).   Google Scholar

[12]

C. Gordon, Problems,, Workshop on Heegaard Splittings, 12 (2007), 401.   Google Scholar

[13]

E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung,, Math. Ann., 112 (1936), 664.   Google Scholar

[14]

K. N. Jones and A. W. Reid, Minimal index torsion-free subgroups of Kleinian groups,, Math. Ann., 310 (1998), 235.   Google Scholar

[15]

M. Kapovich, Hyperbolic manifolds and discrete groups,, Progress in Mathematics, 183 (2001).   Google Scholar

[16]

L. Keen and C. Series, The Riley slice of Schottky space,, Proc. London Math. Soc., 69 (1994), 72.   Google Scholar

[17]

Y. Komori and C. Series, The Riley slice revised,, in, 1 (1999), 303.   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.   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.  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, (1977).   Google Scholar

[29]

G. McShane, "A Remarkable Identity for Lengths of Curves,", Ph. D. Thesis, (1991).   Google Scholar

[30]

G. McShane, Simple geodesics and a series constant over Teichmuller space,, Invent. Math., 132 (1998), 607.   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.   Google Scholar

[32]

M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces,, Invent. Math., 167 (2007), 179.   Google Scholar

[33]

B. B. Newman, Some results on one-relator groups,, Bull. Amer. Math. Soc., 74 (1968), 568.   Google Scholar

[34]

T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$-bridge link groups,, Geom. Topol. Monogr., 14 (2008), 417.   Google Scholar

[35]

R. Riley, Parabolic representations of knot groups. I,, Proc. London Math. Soc., 24 (1972), 217.   Google Scholar

[36]

R. Riley, Algebra for Heckoid groups,, Trans. Amer. Math. Soc., 334 (1992), 389.  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.   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.   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.   Google Scholar

[41]

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

show all references

References:
[1]

C. Adams, Hyperbolic 3-manifolds with two generators,, Comm. Anal. Geom., 4 (1996), 181.   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, (2002).   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, 355 (2004), 21.   Google Scholar

[4]

H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups,, Proceedings of the Workshop, 329 (2006), 151.   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 (2007).   Google Scholar

[6]

M. Boileau and J. Porti, Geometrization of 3-orbifolds of cyclic type,, Appendix A by Michael Heusener and Porti, (2001).   Google Scholar

[7]

M. Boileau and B. Zimmermann, The $\pi$-orbifold group of a link,, Math. Z., 200 (1989), 187.   Google Scholar

[8]

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

[9]

B. H. Bowditch, Markoff triples and quasifuchsian groups,, Proc. London Math. Soc., 77 (1998), 697.   Google Scholar

[10]

B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles,, Topology, 36 (1997), 325.   Google Scholar

[11]

D. Cooper, C. D. Hodgson and S. P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds,, MSJ Memoirs, 5 (2000).   Google Scholar

[12]

C. Gordon, Problems,, Workshop on Heegaard Splittings, 12 (2007), 401.   Google Scholar

[13]

E. Hecke, Über die Bestimung Dirichletscher Reihen durch ihre Funktionalgleichung,, Math. Ann., 112 (1936), 664.   Google Scholar

[14]

K. N. Jones and A. W. Reid, Minimal index torsion-free subgroups of Kleinian groups,, Math. Ann., 310 (1998), 235.   Google Scholar

[15]

M. Kapovich, Hyperbolic manifolds and discrete groups,, Progress in Mathematics, 183 (2001).   Google Scholar

[16]

L. Keen and C. Series, The Riley slice of Schottky space,, Proc. London Math. Soc., 69 (1994), 72.   Google Scholar

[17]

Y. Komori and C. Series, The Riley slice revised,, in, 1 (1999), 303.   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.   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.  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, (1977).   Google Scholar

[29]

G. McShane, "A Remarkable Identity for Lengths of Curves,", Ph. D. Thesis, (1991).   Google Scholar

[30]

G. McShane, Simple geodesics and a series constant over Teichmuller space,, Invent. Math., 132 (1998), 607.   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.   Google Scholar

[32]

M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces,, Invent. Math., 167 (2007), 179.   Google Scholar

[33]

B. B. Newman, Some results on one-relator groups,, Bull. Amer. Math. Soc., 74 (1968), 568.   Google Scholar

[34]

T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between $2$-bridge link groups,, Geom. Topol. Monogr., 14 (2008), 417.   Google Scholar

[35]

R. Riley, Parabolic representations of knot groups. I,, Proc. London Math. Soc., 24 (1972), 217.   Google Scholar

[36]

R. Riley, Algebra for Heckoid groups,, Trans. Amer. Math. Soc., 334 (1992), 389.  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.   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.   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.   Google Scholar

[41]

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

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