Abstract
In this paper, we consider the natural action of \(\textrm{SL}(3, \mathbb {H})\) on the quaternionic projective space \( {\mathbb {P}}_{\mathbb {H}}^2\). Under this action, we investigate limit sets for cyclic subgroups of \(\textrm{SL}(3, \mathbb {H})\). We compute two types of limit sets, which were introduced by Kulkarni and Conze-Guivarc’h, respectively.
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References
Abels, H., Margulis, G.A., Soĭfer, G.A.: Semigroups containing proximal linear maps. Israel J. Math. 91(1–3), 1–30 (1995)
Barrera, W., Cano, A., Navarrete, J.: The limit set for discrete subgroups of \(\rm PSL (3, {\mathbb{C} })\). Math. Proc. Cambridge Philos. Soc. 150(1), 129–146 (2011)
Barrera, W., Cano, A., Navarrete, J.: On the number of lines in the limit set for discrete subgroups of \(\rm PSL (3,{\mathbb{C} })\). Pacific J. Math. 281(1), 17–49 (2016)
Barrera, W., Gonzalez-Urquiza, A., Navarrete, J.P.: Duality of the Kulkarni Limit Set for Subgroups of \({{\rm PSL}}(3, {\mathbb{C} })\). Bull. Braz. Math. Soc. (N.S.) 49, 261–277 (2018)
Maskit, B.: Kleinian groups. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287. Springer-Verlag, Berlin, (1988)
Conze, J.-P., Guivarc’h, Y., (200) Limit sets of groups of linear transformations. Ergodic theory and harmonic analysis (Mumbai,: Sankhyā Ser. A 62, 367–385 (1999)
Cano, A., Loeza, L., Ucan-Puc, A.: Projective cyclic groups in higher dimensions. Linear Algebra Appl. 531, 169–209 (2017)
Cano, A., Navarrete, J.P., Seade, J.: Complex Kleinian groups. Progress in Mathematics, 303. Birkhäuser/Springer Basel AG, Basel, (2013)
Cano, A., Seade, J.: On the equicontinuity region of discrete subgroups of \({\rm PU}(1, n)\). J. Geom. Anal. 20(2), 291–305 (2010)
Cano, A., Seade, J.: On discrete groups of automorphisms of \({\mathbb{P} }^2_{\mathbb{C} }\). Geom. Dedicata 168, 9–60 (2014)
Hasselblatt, B., Katok, A.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications (1995)
Kulkarni, R.S.: Groups with domains of discontinuity. Math. Ann. 237(3), 253–272 (1978)
Marden, A.: Hyperbolic manifolds. An introduction in 2 and 3 dimensions. Cambridge University Press, Cambridge, (2016)
McMullen, C.T.: Complex dynamics and renormalization. Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, (1994)
Myrberg, P.J.: Untersuchungen Über die Automorphen Funktionen Beliebig Vieler Variablen. Acta Math. 46(3–4), 215–336 (1925)
Navarrete, J.P.: On the Limit Set of Discrete Subgroups of \(\rm PU (2,1)\). Geom. Dedicata 122, 1–13 (2006)
Navarrete, J.P.: The trace function and complex Kleinian groups in \({\mathbb{P} }^{2}_{\mathbb{C} }\). Internat. J. Math. 19(7), 865–890 (2008)
Rodman, L.: Topics in quaternion linear algebra. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2014)
Seade, J., Ucan-Puc, A.: Comparison of limit sets for the action of Kleinian groups in \({\mathbb{C}\mathbb{P}}^{n}\). preprint, arXiv:2108.09791
Seade, J., Verjovsky, A.: Actions of discrete groups on complex projective spaces. Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), 155–178, Contemp. Math., 269, Amer. Math. Soc., Providence, RI, (2001)
Seade, J., Verjovsky, A.: Higher dimensional complex Kleinian groups. Math. Ann. 322(2), 279–300 (2002)
Seade, J., Verjovsky, A.: Complex Schottky groups. Geometric methods in dynamics. II. Astérisque No. 287, 251–272 (2003)
Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
Acknowledgements
We are thankful to José Seade and Angel Cano for their comments and suggestions on this paper. The possibility of generalising this paper to arbitrary dimensions was kindly pointed out by Seade. We hope to return to that in a future work. Gongopadhyay is partially supported by the SERB core research grant CRG/2022/003680. Lohan acknowledges full support from the CSIR SRF grant, file No.: 09/947(0113)/ 2019-EMR-I, during the course of this work.
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Dutta, S., Gongopadhyay, K. & Lohan, T. Limit sets of cyclic quaternionic Kleinian groups. Geom Dedicata 217, 61 (2023). https://doi.org/10.1007/s10711-023-00797-9
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DOI: https://doi.org/10.1007/s10711-023-00797-9