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McShane's identity in rank one symmetric spaces

Published online by Cambridge University Press:  15 April 2014

INKANG KIM ,
JOONHYUNG KIM  and
SER PEOW TAN
INKANG KIM
Affiliation:
School of Mathematics, KIAS, Seoul, 130-722, Korea. e-mail: inkang@kias.re.kr
JOONHYUNG KIM
Affiliation:
Department of Mathematics Education, Hannam University, Daejon 306-791, Republic of Korea. e-mail: calvary@snu.ac.kr
SER PEOW TAN
Affiliation:
Department of Mathematics, National University of Singapore, Block S17, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore. e-mail: mattansp@nus.edu.sg
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Abstract

We study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this class of identities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 1 , July 2014 , pp. 113 - 137
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

REFERENCES

[1]Akiyoshi, H., Miyachi, H. and Sakuma, M.Variations of McShane's identity for punctured surface groups, In Spaces of Kleinian Groups. London Math. Soc. Lecture Note series. vol 329, Cambridge Univ. Press. Cambridge, 2006, 151–185.Google Scholar
[2]Birman, J. and Series, C.Geodesics with bounded intersection are sparse. Topology 24 (1985), 217–225.CrossRefGoogle Scholar
[3]Bourdon, M.Structure conforme au bord et flot géodésique d'un CAT(-1)-espace. Enseign. Math. (2) 41 (1995), no. 1–2, 63–102.Google Scholar
[4]Bowditch, B. H.Geometrical finiteness with variable negative curvature. Duke Math. J. 77 (1995), no. 1, 229–274.Google Scholar
[5]Bowditch, B. H.A proof of McShane's identity via Markoff triples. Bull. London Math. Soc. 28 (1996), no. 1, 73–78.Google Scholar
[6]Bowditch, B. H.A variation of McShane's identity for once-punctured torus bundles. Topology 36 (1997), no. 2, 325–334.CrossRefGoogle Scholar
[7]Falbel, E.Geometric structures associated to triangulations as fixed point sets of involutions. Topology Appl. 154 (2007), 1041–1052.Google Scholar
[8]Fu, X., Li, L. and Wang, X.A characterization of Fuchsian groups acting on complex hyperbolic spaces. Czechoslovak Math. J. 62 (137) (2012), 517–525.Google Scholar
[9]Goldman, W. M.Complex hyperbolic Geometry (Oxford University Press, 1999).CrossRefGoogle Scholar
[10]Kim, I.Marked length Rigidity of rank one symmetric spaces and their product. Topology 40 (2001), no. 6, 1295–1323.CrossRefGoogle Scholar
[11]Koranyi, A. and Reimann, H. M.The complex cross-ratio on the Heisenberg groups. L'Enseign. Math. 33 (1987), 291–300.Google Scholar
[12]Labourie, F. and McShane, G.Cross ratios and identities for Higher Teichmüller–Thurston Theory. Duke Math. 148 (9) (2009), 279–345.Google Scholar
[13]McShane, G. A remarkable identity for lengths of curves. Ph.D. Thesis. University of Warwick (1991).Google Scholar
[14]McShane, G.Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998), no. 3, 607–632.CrossRefGoogle Scholar
[15]Mirzakhani, M.Simple geodesics and Weil–Peterson volumes of moduli spaces of bordered Riemann surfaces. Invent Math. 167 (1) (2007), 179–222.Google Scholar
[16]Mirzakhani, M.Weil–Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc. 20 (2007), no. 1, 1–23.Google Scholar
[17]Mirzakhani, M.Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. of Math. (2) 168 (2008), no. 1, 97–125.CrossRefGoogle Scholar
[18]Mostow, G. D.Strong rigidity of locally symmetric spaces.Ann. Math. Stud., vol. 78. (Princeton University Press, Princeton, NJ, 1973).Google Scholar
[19]Parker, J. R., Platis, I. D.. Complex Hyperbolic Fenchel–Nielsen coordinates. Topology 47 (2008), no. 2, 101–135.Google Scholar
[20]Platis, I. D.. Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1, arXiv:1208.5171v2 [math.DG].Google Scholar
[21]Tan, S. P., Wong, Y. L. and Zhang, Y.Generalizations of McShane's identity to hyperbolic cone-surfaces. J. Differential Geom. 72 (2006), no. 1, 73–112.Google Scholar
[22]Tan, S. P., Wong, Y. L. and Zhang, Y.McShane's identity for classical Schottky groups. Pacific J. Math. 237 (2008), no. 1, 183–200.Google Scholar
[23]Tan, S. P., Wong, Y. L. and Zhang, Y.Generalized Markoff maps and McShane's identity. Adv. Math. 217 (2008), 761–813.Google Scholar
[24]Tan, S. P., Wong, Y. L. and Zhang, Y.Delambre–Gauss formulas for augmented, right-angled hexagons in hyperbolic 4-space. Adv. Math. 230 (2012), no. 3, 927–956.Google Scholar