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On Fenchel–Nielsen Coordinates of Surface Group Representations into SU(3,1)

Published online by Cambridge University Press:  01 March 2017

KRISHNENDU GONGOPADHYAY  and
SHIV PARSAD
KRISHNENDU GONGOPADHYAY
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India. e-mail: [email protected], [email protected]
SHIV PARSAD
Affiliation:
Indian Institute of Science Education and Research (IISER) Bhopal, Bhopal Bypass Road, Bhauri Bhopal 462 066 Madhya Pradesh, India. e-mail: [email protected]
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Abstract

Let Σg be a compact, connected, orientable surface of genus g ≥ 2. We ask for a parametrisation of the discrete, faithful, totally loxodromic representations in the deformation space Hom(π1(Σg), SU(3, 1))/SU(3, 1). We show that such a representation, under some hypothesis, can be determined by 30g − 30 real parameters.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 1 - 23
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Cao, W. S. and Gongopadhyay, K. Commuting isometries of the complex hyperbolic space. Proc. Amer. Math. Soc. 139 (2011), no. 9, 3317–3326.CrossRefGoogle Scholar
[2] Chen, S. S. and Greenberg, L. Hyperbolic Spaces. Contributions to Analysis. (Academic Press. New York, 1974), 49–87.Google Scholar
[3] Cunha, H. and Gusevskii, N. On the moduli space of quadruples of points in the boundary of complex hyperbolic space. Transform. Groups 15 (2010), no. 2, 261–283.Google Scholar
[4]Drensky, V. and Sadikova, L. Generators of invariants of two 4 × 4 matrices. C. R. Acad. Bulgare Sci. 59 (5) (2006), 477–484.Google Scholar
[5] Đoković, D. Poincaré series of some pure and mixed trace algebras of two generic matrices. J. Algebra 309 (2) (2007), 654–671.CrossRefGoogle Scholar
[6] Falbel, E. Geometric structures associated to triangulations as fixed point sets of involutions. Topology Appl. 154, No. 6, (2007), 1041–1052.Google Scholar
[7] Fricke, R. Über die Theorie der automorphen Modulgrupper. Nachr. Akad.Wiss. Göttingen (1896), 91–101.Google Scholar
[8] Goldman, W. M. Complex Hyperbolic Geometry (Oxford University Press, 1999).Google Scholar
[9] Goldman, W. M. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In: Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13 (Eur. Math. Soc., Zürich, 2009), 611–684.CrossRefGoogle Scholar
[10] Goldman, W. M. Convex real projective structures on compact surfaces. J. Differential Geom. 31 (1990) 791–845.Google Scholar
[11] Gongopadhyay, K. Algebraic characterization of isometries of the complex and the quaternionic hyperbolic 3-spaces. Proc. Amer. Math. Soc. 141 (2013), 1017–1027.Google Scholar
[12] Gongopadhyay, K. and Lawton, S. Invariants of pairs in SL(4, ${\mathbb C}$) and SU(3, 1). To appear in Proc. Amer. Math. Soc.Google Scholar
[13] Gongopadhyay, K. and Parker, J. R. Reversible complex hyperbolic isometries. Linear Algebra Appl. 438 (2013), 2728–2739.CrossRefGoogle Scholar
[14] Gongopadhyay, K., Parker, J. R. and Parsad, S. On the classification of unitary matrices. Osaka J. Math. 52, no. 4 (2015), 959–993.Google Scholar
[15] Khoi, V. T. On the SU(2,1) representation space of the Brieskorn homology spheres. J. Math. Sci. Univ. Tokyo 14 (2007), no. 4, 499–510.Google Scholar
[16] Kourouniotis, C. Complex length coordinates for quasi-Fuchsian groups. Mathematika 41 (1994), 173–188.Google Scholar
[17] Lawton, S. Generators, relations and symmetries in pairs of 3 ×3 unimodular matrices. J. Algebra 313 (2007) 782–801.CrossRefGoogle Scholar
[18] Parker, J. R. Traces in complex hyperbolic geometry. Geometry, Topology and Dynamics of Character Varieties. Lecture Notes Series 23, Institute for Mathematical Sciences, National University of Singapore (World Scientific Publishing Co. 2012), 191–245.Google Scholar
[19] Parker, J. R. and Platis, I. D. Complex hyperbolic Fenchel–Nielsen coordinates. Topology 47 (2008) 101–135.Google Scholar
[20] Parker, J. R. and Platis, I. D. Complex hyperbolic quasi-Fuchsian groups. Geometry of Riemann Surfaces. London Math. Soc. Lecture Notes 368 (2010), 309–355.Google Scholar
[21] Platis, I. D. Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1. Geom. Dedicata 169 (2014), 187–208.Google Scholar
[22] Schwartz, R. E. Complex hyperbolic triangle groups. Proceedings of the International Congress of Mathematicians (2002) Volume 1:2 Invited Lectures, 339–350.Google Scholar
[23] Strubel, T. Fenchel–Nielsen coordinates for maximal representations. Geom. Dedicata DOI 10.1007/s10711-014-9959-1.Google Scholar
[24] Tan, S. P. Complex Fenchel–Nielsen coordinates for quasi-Fuchsian structures. Internat. J. Math. 5 (1994), 239–251.CrossRefGoogle Scholar
[25] Wen, Z.-X. Relations polynomiales entre les traces de produits de matrices. C. R. Acad. Sci. Paris 318 (1994), 99–104.Google Scholar
[26] Will, P. Groupes libres, groupes triangulaires et tore épointé dans PU(2, 1). Ph.D. thesis. University of Paris VI.Google Scholar
[27] Will, P. Traces, cross-ratios and 2-generator subgroups of SU(2, 1). Canad. J. Math. 61 (2009), 1407–1436.Google Scholar
[28] Will, P. Two Generator groups acting on the complex hyperbolic plane. arXiv 1510.01500, to appear as a chapter of Volume VI of the Handbook of Teichmüller theory.Google Scholar
[29] Vogt, H. Sur les invariants fondamentaux deséquations différentielles linéaires du second ordre. Ann. Sci. E. N. S. 3eme Série, Tome VI (1889), supplement S.3–S.70.Google Scholar