Abstract
Susskind’s conjecture states that for subgroups \(\Phi \) and \(\Theta \) of a Kleinian group \(\Gamma \) acting on \({\mathbb H}^n\), we have that \(\Lambda _c(\Phi )\cap \Lambda _c (\Theta )\subset \Lambda (\Phi \cap \Theta )\), where \(\Lambda _c(\Phi )\) is the set of conical limit points of \(\Phi \) and \(\Lambda (\Phi )\) is the limit set of \(\Phi \). We show that Susskind’s conjecture largely holds for purely loxodromic Kleinian groups and we present two examples to illustrate that Susskind’s conjecture is nearly optimal.
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References
Agol, I.: Tameness of hyperbolic 3-manifolds. Preprint, arXiv:math/0405568
Anderson, J.W.: Intersections of analytically and geometrically finite subgroups of Kleinian groups. Trans. Am. Math. Soc. 343, 87–98 (1994)
Anderson, J.W.: Intersections of topologically tame subgroups of Kleinian groups. J. d’Anal. Math. 65, 77–94 (1995)
Anderson, J.W.: The limit set intersection theorem for finitely generated Kleinian groups. Math. Res. Lett. 3, 675–692 (1996)
Apanasov, B.N.: Conformal geometry of discrete groups and manifolds. de Gruyter Expositions in Mathematics 32. de Gruyter, Berlin (2000)
Bishop, C.J.: On a theorem of Beardon and Maskit. Ann. Acad. Sci. Fenn. 21, 383–388 (1996)
Bishop, C.J., Jones, P.W.: Hausdorff dimension and Kleinian groups. Acta Math. 179, 1–39 (1997)
Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Am. Math. Soc. 19, 385–446 (2006)
Kapovich, M.: Hyperbolic manifolds and discrete groups, Progress in Mathematics 183. Birkhäuser Boston Inc., Boston (2001)
Maskit, B.: Kleinian groups, Grundlehren der Mathematischen Wissenschaften 287. Springer, Berlin (1988)
Maskit, B.: Intersections of component subgroups of Kleinian groups. In: Greenberg, L. (ed) Discontinuous groups and Riemann surfaces, Annals of Mathematics Studies 79, pp. 349–367. Princeton University Press, Princeton (1974)
Matsuzaki, K., Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs, Oxford University Press, New York (1998)
Ratcliffe, J.G.: Foundations of hyperbolic manifolds, 2nd edn. Graduate Texts in Mathematics 149. Springer, New York (2006)
Soma, T.: Function groups in Kleinian groups. Math. Ann. 292, 181–190 (1992)
Susskind, P.: Kleinian groups with intersecting limit sets. J. d’Anal. Math. 52, 26–38 (1989)
Susskind, P.: On Kleinian groups with intersecting limit sets. PhD Thesis, SUNY Stony Brook (1982)
Susskind, P., Swarup, G.A.: Limits set of geometrically finite Kleinian groups. Am. J. Math. 114, 233–250 (1992)
Yang, W.: Limit sets of relatively hyperbolic groups. Geom. Dedicat. 156, 1–12 (2012)
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Communicated by Gaven Martin.
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Anderson, J.W. Limit Set Intersection Theorems for Kleinian Groups and a Conjecture of Susskind. Comput. Methods Funct. Theory 14, 453–464 (2014). https://doi.org/10.1007/s40315-014-0078-7
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DOI: https://doi.org/10.1007/s40315-014-0078-7