Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T04:59:09.920Z Has data issue: false hasContentIssue false
  • English
  • Français

TEST MAP AND DISCRETENESS IN SL(2, ℍ)

Part of: Other groups of matrices Real and complex geometry

Published online by Cambridge University Press:  07 August 2018

KRISHNENDU GONGOPADHYAY Open the ORCID record for KRISHNENDU GONGOPADHYAY [Opens in a new window] ,
ABHISHEK MUKHERJEE  and
SUJIT KUMAR SARDAR
KRISHNENDU GONGOPADHYAY*
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab 140306, India e-mail: [email protected], [email protected]
ABHISHEK MUKHERJEE*
Affiliation:
Kalna College, Kalna, Dist. Burdwan 713409, West BengalDepartment of Mathematics, Jadavpur University, Jadavpur 700032, Kolkata e-mail: [email protected]
SUJIT KUMAR SARDAR*
Affiliation:
Department of Mathematics, Jadavpur University, Jadavpur 700032, Kolkata e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

MSC classification

Primary: 20H10: Fuchsian groups and their generalizations
Secondary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 20H25: Other matrix groups over rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 523 - 533
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

Abikoff, W. and Haas, A., Nondiscrete groups of hyperbolic motions, Bull. London Math. Soc. 22 (3) (1990), 233–238.CrossRefGoogle Scholar
Bisi, C. and Gentili, G., Möbius transformations and the Poincaré distance in the quaternionic setting, Indiana Univ. Math. J. 58 (6) (2009), 2729–2764.CrossRefGoogle Scholar
Beardon, A. F., The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91 (Springer-Verlag, New York, 1995), Corrected reprint of the 1983 original.Google Scholar
Cao, W., Discreteness criterion in SL(2, ℂ) by a test map, Osaka J. Math. 49 (4) (2012), 901–907.Google Scholar
Cao, W., On the classification of four-dimensional Möbius transformations, Proc. Edinb. Math. Soc., II. Ser. 50 (1) (2007), 49–62.CrossRefGoogle Scholar
Chen, S. S. and Greenberg, L., Hyperbolic spaces, in Contributions to analysis (a collection of papers dedicated to Lipman Bers) (Academic Press, New York, 1974), 49–87.Google Scholar
Chen, M., Discreteness and convergence of Möbius groups, Geom. Dedicata 104 (2004), 61–69.CrossRefGoogle Scholar
Foreman, B., Conjugacy invariants of Sl(2, ℍ), Linear Algebra Appl. 381 (2004), 25–35.CrossRefGoogle Scholar
Fang, A. and Nai, B., On the discreteness and convergence in n-dimensional Möbius groups, J. London Math. Soc. II. Ser. 61 (3) (2000), 761–773.Google Scholar
Gongopadhyay, K. and Mukherjee, A., Extremality of quaternionic Jørgensen inequality, Hiroshima Math. J. 47 (2) (2017), 113–137.CrossRefGoogle Scholar
Gongopadhyay, K., Algebraic characterization of the isometries of the hyperbolic 5-space, Geom. Dedicata 144 (2010), 157–170.CrossRefGoogle Scholar
Kellerhals, R., Quaternions and some global properties of hyperbolic 5-manifolds, Canad. J. Math. 55 (5) (2003), 1080–1099.CrossRefGoogle Scholar
Li, L.-L. and Wang, X.-T., Discreteness criteria for Möbius groups acting on ℝn II, Bull. Aust. Math. Soc. 80 (2) (2009) 275–290.CrossRefGoogle Scholar
Parker, J. R., Hyperbolic spaces, Jyväskylä Lectures in Mathematics, vol. 2 (University of Jyväskylä, Finland, 2008).Google Scholar
Parker, J. R. and Short, I., Conjugacy classification of quaternionic Möbius transformations, Comput. Methods Funct. Theory 9 (1) (2009), 13–25.CrossRefGoogle Scholar
Ratcliffe, J. G., Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149 (Springer-Verlag, New York, 2nd edition, 2005).Google Scholar
Tukia, P. and Wang, X., Discreteness of subgroups of SL(2,C) containing elliptic elements, Math. Scand. 91 (2002), 214–220.CrossRefGoogle Scholar
Wang, X., Li, L. and Cao, W., Discreteness criteria for Möbius groups acting on ℝn, Israel J. Math. 150 (2005), 357–368.CrossRefGoogle Scholar
Waterman, P., Möbius groups in several dimensions, Adv. Math. 101 (1993), 87–113.CrossRefGoogle Scholar
Wilker, J. B., The quaternion formalism for Möbius groups in four or fewer dimensions, Linear Algebra Appl. 190 (1993), 99–136.CrossRefGoogle Scholar
Yang, S., Elliptic elements in Möbius groups, Israel J. Math. 172 (2009), 309–315.CrossRefGoogle Scholar
Yang, S., Test maps and discrete groups in SL(2, C), Osaka J. Math. 46 (2) (2009), 403–409.Google Scholar
Yang, S. and Zhao, T., Test maps and discrete groups in SL(2, ℂ) II, Glasg. Math. J. 56 (1) (2014), 53–56.CrossRefGoogle Scholar
Yang, S., On geometric convergence of discrete groups, Czech. Math. J. 64 (139) (2014), 305–310.CrossRefGoogle Scholar
Yang, S. and Zhao, T., Conjugacy class and discreteness in SL(2, ℂ), Osaka J. Math. 53 (4) (2016), 1047–1053.Google Scholar