The ${\mathrm {SL}(2,{\mathbb C})}$ character variety of a one-holed torus
Authors:
Ser Peow Tan, Yan Loi Wong and Ying Zhang
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 103-110
MSC (2000):
Primary 57M50
DOI:
https://doi.org/10.1090/S1079-6762-05-00153-8
Published electronically:
December 23, 2005
MathSciNet review:
2191691
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Abstract: In this note we announce several results concerning the ${\mathrm {SL}(2,{\mathbb C})}$ character variety ${\mathcal X}$ of a one-holed torus. We give a description of the largest open subset ${\mathcal X}_{BQ}$ of ${\mathcal X}$ on which the mapping class group $\Gamma$ acts properly discontinuously, in terms of two very simple conditions, and show that a series identity generalizing McShane’s identity for the punctured torus holds for all characters in this subset. We also give variations of the McShane-Bowditch identities for characters fixed by an Anosov element of $\Gamma$ with applications to closed hyperbolic three-manifolds. Finally we give a definition of end invariants for ${\mathrm {SL}(2,{\mathbb C})}$ characters and give a partial classification of the set of end invariants of a character in ${\mathcal X}$.
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akiyoshi-miyachi-sakuma2004preprint Hirotaka Akiyoshi, Hideki Miyachi, and Makoto Sakuma, Variations of McShane’s identity for punctured surface groups, London Mathematical Society Lecture Notes, Y. Minsky, M. Sakuma & C. Series (Eds.), Cambridge University Press (to appear).
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goldman-tan-zhang2005 William M. Goldman, Ser Peow Tan, and Ying Zhang, The ${\mathrm {SL}(2,{\mathbb C})}$ character variety of the four-holed sphere, in preparation.
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mcshane1991thesis Greg McShane, A remarkable identity for lengths of curves, Ph.D. Thesis, University of Warwick, 1991.
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mirzakhani2004preprint Maryam Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, preprint.
mirzakhani22004preprint Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, preprint.
mirzakhani20043preprint Maryam Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, preprint.
tan-wong-zhang2004cone-surfaces Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Generalizations of McShane’s identity to hyperbolic cone-surfaces, arXiv:math.GT/0404226, to appear, J. Differential Geom.
tan-wong-zhang2004schottky Ser Peow Tan, Yan Loi Wong, and Ying Zhang, McShane’s identity for classical Schottky groups, arXiv:math.GT/0411628.
tan-wong-zhang2004necsuf Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Necessary and sufficient conditions for McShane’s identity and variations, arXiv:math.GT/0411184.
tan-wong-zhang2004gMm Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Generalized Markoff maps and McShane’s identity, arXiv:math.GT/0502464.
tan-wong-zhang2004endinvariants Ser Peow Tan, Yan Loi Wong, and Ying Zhang, End invariants for ${\mathrm {SL}(2,{\mathbb C})}$ characters of the one-holed torus, arXiv:math.GT/0511621.
thurston1978notes William P. Thurston, The geometry and topology of $3$-manifolds, Lecture Notes, Princeton University, 1977/78.
thurston19am William P. Thurston, Hyperbolic structures on $3$-manifolds II: surface groups and $3$-manifolds that fiber over the circle, arXiv:math.GT/9801045 v1.
zhang2004thesis Ying Zhang, Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane’s identity, Ph.D. Thesis, National University of Singapore, 2004.
akiyoshi-miyachi-sakuma2004cm355 Hirotaka Akiyoshi, Hideki Miyachi, and Makoto Sakuma, A refinement of McShane’s identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III. The Ahlfors-Bers Colloquium, Oct. 2001, Univ. of Connecticut at Storrs, W. Abikoff, A. Haas (Eds.), Contemporary Mathematics 355, pp. 21–40, American Mathematical Society, 2004.
akiyoshi-miyachi-sakuma2004preprint Hirotaka Akiyoshi, Hideki Miyachi, and Makoto Sakuma, Variations of McShane’s identity for punctured surface groups, London Mathematical Society Lecture Notes, Y. Minsky, M. Sakuma & C. Series (Eds.), Cambridge University Press (to appear).
bowditch1996blms Brian H. Bowditch, A proof of McShane’s identity via Markoff triples, Bull. London Math. Soc. 28 (1996), no. 1, 73–78.
bowditch1997t Brian H. Bowditch, A variation of McShane’s identity for once-punctured torus bundles, Topology 36 (1997), no. 2, 325–334.
bowditch1998plms Brian H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3) 77 (1998), no. 3, 697–736.
goldmanAM1997 William M. Goldman, Ergodic theory on moduli spaces, Ann. Math. 146 (1997), 475–507.
goldmanGT2003 William M. Goldman, The modular group action on real $\textrm {SL}(2)$-characters of a one-holed torus, Geom. Topol. 7 (2003), 443–486.
goldman-tan-zhang2005 William M. Goldman, Ser Peow Tan, and Ying Zhang, The ${\mathrm {SL}(2,{\mathbb C})}$ character variety of the four-holed sphere, in preparation.
grothendieck1997 Alexandre Grothendieck, Esquisse d’un programme, London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, Cambridge Univ. Press, Cambridge, 1997, pp. 5–48.
luo1999 Feng Luo, Characters of $\textrm {SL}(2)$ representations of groups, J. Differential Geom. 53 (1999), no. 3, 575–626.
mcshane1991thesis Greg McShane, A remarkable identity for lengths of curves, Ph.D. Thesis, University of Warwick, 1991.
mcshane1998im Greg McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998), no. 3, 607–632.
mirzakhani2004preprint Maryam Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, preprint.
mirzakhani22004preprint Maryam Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, preprint.
mirzakhani20043preprint Maryam Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves, preprint.
tan-wong-zhang2004cone-surfaces Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Generalizations of McShane’s identity to hyperbolic cone-surfaces, arXiv:math.GT/0404226, to appear, J. Differential Geom.
tan-wong-zhang2004schottky Ser Peow Tan, Yan Loi Wong, and Ying Zhang, McShane’s identity for classical Schottky groups, arXiv:math.GT/0411628.
tan-wong-zhang2004necsuf Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Necessary and sufficient conditions for McShane’s identity and variations, arXiv:math.GT/0411184.
tan-wong-zhang2004gMm Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Generalized Markoff maps and McShane’s identity, arXiv:math.GT/0502464.
tan-wong-zhang2004endinvariants Ser Peow Tan, Yan Loi Wong, and Ying Zhang, End invariants for ${\mathrm {SL}(2,{\mathbb C})}$ characters of the one-holed torus, arXiv:math.GT/0511621.
thurston1978notes William P. Thurston, The geometry and topology of $3$-manifolds, Lecture Notes, Princeton University, 1977/78.
thurston19am William P. Thurston, Hyperbolic structures on $3$-manifolds II: surface groups and $3$-manifolds that fiber over the circle, arXiv:math.GT/9801045 v1.
zhang2004thesis Ying Zhang, Hyperbolic cone-surfaces, generalized Markoff maps, Schottky groups and McShane’s identity, Ph.D. Thesis, National University of Singapore, 2004.
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Additional Information
Ser Peow Tan
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email:
mattansp@nus.edu.sg
Yan Loi Wong
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email:
matwyl@nus.edu.sg
Ying Zhang
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Address at time of publication:
Department of Mathematics, Yangzhou University, Yangzhou 225002, P. R. China
Email:
yingzhang@alumni.nus.edu.sg
Received by editor(s):
September 6, 2005
Published electronically:
December 23, 2005
Additional Notes:
The authors are partially supported by the National University of Singapore academic research grant R-146-000-056-112. The third author is also partially supported by the National Key Basic Research Fund (China) G1999075104.
Communicated by:
Walter Neumann
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
