Jeon, Woojin; Kim, Inkang; Ohshika, Ken’ichi Primitive stable representations of free Kleinian groups. (English) Zbl 1361.57025 Isr. J. Math. 199, Part B, 841-866 (2014). Let \(F\) be a non-abelian free group of rank \(n\). The authors give a complete criterion for a discrete faithful representation \(\rho : F_{n} \to \text{PSL} (2, \mathbb C)\) to be primitive stable. Theorem 1.1 states that if \(\rho\) is a discrete faithful representation of \(F\) without parabolics then \(\rho\) is primitive stable. Theorem 1.2 states the following. Let \(\rho\) be a discrete, faithful and geometrically infinite representation with parabolics such that the non-cuspidal part \(M_{0}\) of \(M = \mathbb H^{3} / \rho (F)\) is the union of the relative compact core \(H\) and finitely many end neighbourhoods \(E_{i}\) facing \(S_{i} \subset \partial H\). Then the representation \(\rho\) is primitive stable if and only if every parabolic curve is disc-busting, and every geometrically infinite end \(E_{i}\) has the ending lamination \(\lambda_{i}\) which is disc-busting on \(\partial H\). These results answer Y. N. Minsky’s conjectures [Isr. J. Math. 193, 47–70 (2013; Zbl 1282.57023)] about geometric conditions on \(\mathbb H^{3} / \rho(F_{n})\) regarding the primitive stability of \(\rho\). Reviewer: Andrei Vesnin (Novosibirsk) Cited in 7 Documents MSC: 57M60 Group actions on manifolds and cell complexes in low dimensions 20F67 Hyperbolic groups and nonpositively curved groups 20E05 Free nonabelian groups Keywords:Kleinian groups; discrete faithful representations; laminations; Masur domain; parabolic curve Citations:Zbl 1282.57023 PDFBibTeX XMLCite \textit{W. Jeon} et al., Isr. J. Math. 199, Part B, 841--866 (2014; Zbl 1361.57025) Full Text: DOI arXiv References: [1] I. Agol, Tameness of hyperbolic 3-manifolds, preprint, arXiv:math.GT/0405568 (2004). [2] M. Bestvina, M. Feighn and M. Handel, The Tits alternative for Out(Fn) I:Dynamics of exponentially-growing automorphisms, Annals of Mathematics 151 (2000), 517–623. · Zbl 0984.20025 [3] B. H. Bowditch, The Cannon-Thurston map for punctured surface groups, Mathematische Zeitschrift 255 (2007), 35–76. · Zbl 1138.57020 [4] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Postive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer, Berlin, 1999. · Zbl 0988.53001 [5] D. Calegari and D. Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, Journal of the American Mathematical Society 19 (2006), 385–446. · Zbl 1090.57010 [6] R. D. Canary, Ends of hyperbolic 3-manifolds, Journal of the American Mathematical Society 6 (1993), 1–35. · Zbl 0810.57006 [7] R. D. Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996), 751–778. · Zbl 0863.57010 [8] R. D. Canary, D. B. A. Epstein and P. Green, Notes on Notes of Thurston, London Mathematical Society Lecture Note Series, Vol. 111, Cambridge University Press, Cambridge, 1987. · Zbl 0612.57009 [9] J. Cannon and W. P. Thurston, Group invariant Peano curves, Geometry and Topology 11 (2007), 1315–1356. · Zbl 1136.57009 [10] A. Casson and S. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, Vol. 9, Cambridge University Press, Cambridge, 1988. · Zbl 0649.57008 [11] A. Casson and D. Long, Algorithmic compression of surface automorphisms, Inventiones Mathematicae 81 (1985), 295–303. · Zbl 0589.57008 [12] S. Das and M. Mj, Addendum to ”Ending laminations and Cannon-Thurston maps: Parabolics”, preprint, arXiv:1002.2090 (2010). [13] W. J. Floyd, Group completions and limit sets of Kleinian groups, Inventiones Mathematicae 57 (1980), 205–218. · Zbl 0428.20022 [14] W. Jeon and I. Kim, Primitive stable representations of geometrically infinite handlebody hyperbolic 3-manifolds, Comptes Rendus Mathématique. Académie des Sciences. Paris 348 (2010), 907–910. · Zbl 1206.57022 [15] E. Klarreich, Semiconjugacies between Kleinian group actions on the Riemann sphere, American Journal of Mathematics 121 (1999), 1031–1078. · Zbl 1011.30035 [16] G. Kleineidam, J. Souto, Algebraic convergence of function groups, Commentarii Mathematici Helvetici 77 (2002), 244–269. · Zbl 1008.30026 [17] C. Lecuire, Plissage des variété hyperboliques de dimension 3, Inventiones Mathematicae 164 (2006), 85–141. · Zbl 1097.57017 [18] C. Lecuire, An extension of the Masur domain, in Spaces of Kleinian Groups, London Mathematical Society Lecture Note Series, Vol. 329, Cambridge University Press, Cambridge, 2006, pp. 49–73 · Zbl 1102.30043 [19] M. Lee, Dynamics on the PSL(2,C)-character variety of a twisted I-bundle, arXiv:1103.3479 · Zbl 1321.57019 [20] A. Lubotzky, Dynamics of Aut(Fn) actions on group presentations and representations, in Geometry, Rigidity and Group Actions, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011, pp. 609–643. · Zbl 1266.20045 [21] B. Maskit, A characterization of Schottky groups, Journal d’Analyse Mathématique 19 (1967), 227–230. · Zbl 0168.06201 [22] H. A. Masur, Measured foliations and handlebodies, Ergodic Theory and Dynamical Systems 6 (1986), 99–116. · Zbl 0628.57010 [23] C. McMullen, Local connectivity, Kleinian groups and geodesics on the blow-up of the torus, Inventiones Mathematicae 97 (2000), 95–127. · Zbl 0672.30017 [24] Y. N. Minsky, On rigidity, limit sets and end invariants of hyperbolic 3-manifolds, Journal of the American Mathematical Society 7 (1994), 539–588. · Zbl 0808.30027 [25] Y. Minsky, On dynamics of Out(F n) on PSL 2(\(\mathbb{C}\)) characters, Israel Journal of Mathemtics 193 (2013), 47–70. · Zbl 1282.57023 [26] M. Mitra, Ending laminations for hyperbolic group extensions, Geometric and Functional Analysis 7 (1997), 379–402. · Zbl 0880.57001 [27] M. Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, Journal of Differential Geometry 48 (1998), 135–164. · Zbl 0906.20023 [28] M. Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology 37 (1998), 527–538. · Zbl 0907.20038 [29] H. Miyachi, Semiconjugacy between the actions of topologically tame Kleinian groups, preprint, (2002). [30] M. Mj, Cannon-Thurston map for Surface groups, preprint, arXiv:math.GT/0511041 (2006). [31] M. Mj, Ending laminations and Cannon-Thurston map, preprint, arXiv:math.GT/0701725 (2008). [32] M. Mj, Cannon-Thurston maps for pared manifolds of bounded geometry, Geometry and Topology 13 (2009), 189–245. · Zbl 1166.57009 [33] M. Mj, Cannon-Thurston maps for Kleinian groups, preprint (2010). · Zbl 1204.57014 [34] K. Ohshika, Ending laminations and boundaries for deformation spaces of Kleinian groups, Journal of the London Mathematical Society 42 (1990), 111–121. · Zbl 0715.30032 [35] J. P. Otal, Courants géodésiques et produits libres, Thèse d’Etat, Université de Paris-Sud, Orsay 1988. [36] J. Souto, Cannon-Thurston maps for thick free groups, preprint (2006). [37] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Annals of Mathematics Studies, Vol. 97, Princeton University Press, Princeton, NJ, 1981, pp. 465–496. · Zbl 0567.58015 [38] W. P. Thurston, Three-dimensional Geometry and Topology, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997. · Zbl 0873.57001 [39] J. H. C. Whitehead, On certain sets of elements in a free group, Proceedings of the London Mathematical Society 41 (1936), 48–56. · Zbl 0013.24801 [40] J. H. C. Whitehead, On equivalent sets of elements in a free group, Annals of Mathematics 37 (1936), 782–800. · Zbl 0015.24804 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.