Electronic Research Announcements

Author(s): Ser Peow Tan; Yan Loi Wong; Ying Zhang
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 103-110.
MSC (2000): Primary 57M50
Posted: December 23, 2005
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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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16.: 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.
17.: Ser Peow Tan, Yan Loi Wong, and Ying Zhang, McShane's identity for classical Schottky groups, arXiv:math.GT/0411628.
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19.: Ser Peow Tan, Yan Loi Wong, and Ying Zhang, Generalized Markoff maps and McShane's identity, arXiv:math.GT/0502464.
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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

DOI: 10.1090/S1079-6762-05-00153-8
PII: S 1079-6762(05)00153-8
Received by editor(s): September 6, 2005
Posted: 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
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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