# zbMATH — the first resource for mathematics

Conical limit points and the Cannon-Thurston map. (English) Zbl 1375.20045
Summary: Let $$G$$ be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space $$Z$$ so that there exists a continuous $$G$$-equivariant map $$i:\partial G\to Z$$, which we call a Cannon-Thurston map. We obtain two characterizations (a dynamical one and a geometric one) of conical limit points in $$Z$$ in terms of their pre-images under the Cannon-Thurston map $$i$$. As an application we prove, under the extra assumption that the action of $$G$$ on $$Z$$ has no accidental parabolics, that if the map $$i$$ is not injective, then there exists a non-conical limit point $$z\in Z$$ with $$| i^{-1}(z)|=1$$. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of word-hyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if $$G$$ is a non-elementary torsion-free word-hyperbolic group, then there exists $$x\in \partial G$$ such that $$x$$ is not a “controlled concentration point” for the action of $$G$$ on $$\partial G$$.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$ 37F40 Geometric limits in holomorphic dynamics 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010) 57M60 Group actions on manifolds and cell complexes in low dimensions
Full Text:
##### References:
 [1] [Ag] I. Agol, Tameness of hyperbolic 3-manifolds, preprint, 2004; arXiv:0405568 [2] Anderson, James W.; Bonfert-Taylor, Petra; Taylor, Edward C., Convergence groups, Hausdorff dimension, and a theorem of Sullivan and Tukia, Geom. Dedicata, 103, 51-67, (2004) · Zbl 1067.30041 [3] Aebischer, Beat; Hong, Sungbok; McCullough, Darryl, Recurrent geodesics and controlled concentration points, Duke Math. J., 75, 3, 759-774, (1994) · Zbl 0810.30034 [4] Baker, O.; Riley, T. R., Cannon-Thurston maps do not always exist, Forum Math. Sigma, 1, e3, 11 pp., (2013) · Zbl 1276.20054 [5] [BRi] O. Baker and T. Riley, Cannon-Thurston maps, subgroup distortion, and hyperbolic hydra, preprint, 2012; arXiv:1209.0815 [6] Beardon, Alan F.; Maskit, Bernard, Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., 132, 1-12, (1974) · Zbl 0277.30017 [7] Bestvina, M.; Feighn, M., A combination theorem for negatively curved groups, J. Differential Geom., 35, 1, 85-101, (1992) · Zbl 0724.57029 [8] Bestvina, Mladen; Handel, Michael, Train tracks and automorphisms of free groups, Ann. of Math. (2), 135, 1, 1-51, (1992) · Zbl 0757.57004 [9] Bestvina, M.; Feighn, M.; Handel, M., Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal., 7, 2, 215-244, (1997) · Zbl 0884.57002 [10] Bestvina, Mladen; Feighn, Mark; Handel, Michael, The Tits alternative for $${\rm Out}(F_n)$$. I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2), 151, 2, 517-623, (2000) · Zbl 0984.20025 [11] Bestvina, Mladen; Feighn, Mark, A hyperbolic $${\rm Out}(F_n)$$-complex, Groups Geom. Dyn., 4, 1, 31-58, (2010) · Zbl 1190.20017 [12] Bestvina, Mladen; Feighn, Mark, Hyperbolicity of the complex of free factors, Adv. Math., 256, 104-155, (2014) · Zbl 1348.20028 [13] Bestvina, Mladen; Reynolds, Patrick, The boundary of the complex of free factors, Duke Math. J., 164, 11, 2213-2251, (2015) · Zbl 1337.20040 [14] Bogopolski, Oleg, Introduction to group theory, EMS Textbooks in Mathematics, x+177 pp., (2008), European Mathematical Society (EMS), Z\"urich · Zbl 1215.20001 [15] Bonahon, Francis, Bouts des vari\'et\'es hyperboliques de dimension $$3$$, Ann. of Math. (2), 124, 1, 71-158, (1986) · Zbl 0671.57008 [16] Bowditch, Brian H., A topological characterisation of hyperbolic groups, J. Amer. Math. Soc., 11, 3, 643-667, (1998) · Zbl 0906.20022 [17] Bowditch, B. H., Convergence groups and configuration spaces. Geometric group theory down under, Canberra, 1996, 23-54, (1999), de Gruyter, Berlin · Zbl 0952.20032 [18] Bowditch, Brian H., The Cannon-Thurston map for punctured-surface groups, Math. Z., 255, 1, 35-76, (2007) · Zbl 1138.57020 [19] Bowditch, B. H., Stacks of hyperbolic spaces and ends of 3-manifolds. Geometry and topology down under, Contemp. Math. 597, 65-138, (2013), Amer. Math. Soc., Providence, RI · Zbl 1297.57044 [20] Bridson, Martin R.; Groves, Daniel, The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Mem. Amer. Math. Soc., 203, 955, xii+152 pp., (2010) · Zbl 1201.20037 [21] Bridson, Martin R.; Haefliger, Andr\'e, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319, xxii+643 pp., (1999), Springer-Verlag, Berlin · Zbl 0988.53001 [22] Brinkmann, P., Hyperbolic automorphisms of free groups, Geom. Funct. Anal., 10, 5, 1071-1089, (2000) · Zbl 0970.20018 [23] Brock, Jeffrey F.; Canary, Richard D.; Minsky, Yair N., The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. (2), 176, 1, 1-149, (2012) · Zbl 1253.57009 [24] Calegari, Danny; Gabai, David, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc., 19, 2, 385-446, (2006) · Zbl 1090.57010 [25] Cannon, James W.; Thurston, William P., Group invariant Peano curves, Geom. Topol., 11, 1315-1355, (2007) · Zbl 1136.57009 [26] Casson, Andrew; Jungreis, Douglas, Convergence groups and Seifert fibered $$3$$-manifolds, Invent. Math., 118, 3, 441-456, (1994) · Zbl 0840.57005 [27] Casson, Andrew J.; Bleiler, Steven A., Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts 9, iv+105 pp., (1988), Cambridge University Press, Cambridge · Zbl 0649.57008 [28] Coulbois, Thierry; Hilion, Arnaud, Botany of irreducible automorphisms of free groups, Pacific J. Math., 256, 2, 291-307, (2012) · Zbl 1259.20031 [29] Coulbois, Thierry; Hilion, Arnaud, Rips induction: index of the dual lamination of an $$\mathbb{R}$$-tree, Groups Geom. Dyn., 8, 1, 97-134, (2014) · Zbl 1336.20033 [30] Coulbois, Thierry; Hilion, Arnaud; Lustig, Martin, Non-unique ergodicity, observers’ topology and the dual algebraic lamination for $$\mathbb{R}$$-trees, Illinois J. Math., 51, 3, 897-911, (2007) · Zbl 1197.20020 [31] Coulbois, Thierry; Hilion, Arnaud; Lustig, Martin, $$\mathbb{R}$$-trees and laminations for free groups. I. Algebraic laminations, J. Lond. Math. Soc. (2), 78, 3, 723-736, (2008) · Zbl 1197.20019 [32] Coulbois, Thierry; Hilion, Arnaud; Lustig, Martin, $$\mathbb{R}$$-trees and laminations for free groups. II. The dual lamination of an $$\mathbb{R}$$-tree, J. Lond. Math. Soc. (2), 78, 3, 737-754, (2008) · Zbl 1198.20023 [33] [DKT] S. Dowdall, I. Kapovich, and S. J. Taylor, Cannon-Thurston maps for hyperbolic free group extensions, Israel J. Math., to appear; arXiv:1506.06974 · Zbl 1361.20030 [34] Fenley, S\'ergio, Ideal boundaries of pseudo-Anosov flows and uniform convergence groups with connections and applications to large scale geometry, Geom. Topol., 16, 1, 1-110, (2012) · Zbl 1279.37026 [35] Floyd, William J., Group completions and limit sets of Kleinian groups, Invent. Math., 57, 3, 205-218, (1980) · Zbl 0428.20022 [36] Freden, Eric M., Negatively curved groups have the convergence property. I, Ann. Acad. Sci. Fenn. Ser. A I Math., 20, 2, 333-348, (1995) · Zbl 0847.20031 [37] Gabai, David, Convergence groups are Fuchsian groups, Ann. of Math. (2), 136, 3, 447-510, (1992) · Zbl 0785.57004 [38] Gerasimov, Victor, Expansive convergence groups are relatively hyperbolic, Geom. Funct. Anal., 19, 1, 137-169, (2009) · Zbl 1226.20037 [39] Gerasimov, Victor, Floyd maps for relatively hyperbolic groups, Geom. Funct. Anal., 22, 5, 1361-1399, (2012) · Zbl 1276.20050 [40] Gerasimov, Victor; Potyagailo, Leonid, Quasi-isometric maps and Floyd boundaries of relatively hyperbolic groups, J. Eur. Math. Soc. (JEMS), 15, 6, 2115-2137, (2013) · Zbl 1292.20047 [41] [GP13] V. Gerasimov and L. Potyagailo, Similar relatively hyperbolic actions of a group, preprint, 2013; arXiv:1305.6649 [42] Gehring, F. W.; Martin, G. J., Discrete quasiconformal groups. I, Proc. London Math. Soc. (3), 55, 2, 331-358, (1987) · Zbl 0628.30027 [43] Handel, Michael; Mosher, Lee, Axes in outer space, Mem. Amer. Math. Soc., 213, 1004, vi+104 pp., (2011) · Zbl 1238.57002 [44] [HM13] M. Handel and L. Mosher, Subgroup decomposition in $$Out(F_n)$$: Introduction and Research Announcement, preprint, 2013; arXiv:1302.2681 [45] Jeon, Woojin; Kim, Inkang; Ohshika, Ken’ichi; Lecuire, Cyril, Primitive stable representations of free Kleinian groups, Israel J. Math., 199, 2, 841-866, (2014) · Zbl 1361.57025 [46] [JO] W. Jeon and K. Ohshika, Measurable rigidity for Kleinian groups, Ergodic Theory and Dynamical Systems, to appear; published online June 2015; DOI: 10.1017/etds.2015.15 [47] Kapovich, Ilya, A non-quasiconvexity embedding theorem for hyperbolic groups, Math. Proc. Cambridge Philos. Soc., 127, 3, 461-486, (1999) · Zbl 0942.20026 [48] Kapovich, Ilya, Algorithmic detectability of iwip automorphisms, Bull. Lond. Math. Soc., 46, 2, 279-290, (2014) · Zbl 1319.20030 [49] Kapovich, Ilya; Benakli, Nadia, Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296, 39-93, (2002), Amer. Math. Soc., Providence, RI · Zbl 1044.20028 [50] Kapovich, Ilya; Lustig, Martin, Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol., 13, 3, 1805-1833, (2009) · Zbl 1194.20046 [51] Kapovich, Ilya; Lustig, Martin, Intersection form, laminations and currents on free groups, Geom. Funct. Anal., 19, 5, 1426-1467, (2010) · Zbl 1242.20052 [52] Kapovich, Ilya; Lustig, Martin, Invariant laminations for irreducible automorphisms of free groups, Q. J. Math., 65, 4, 1241-1275, (2014) · Zbl 1348.20035 [53] Kapovich, Ilya; Lustig, Martin, Cannon-Thurston fibers for iwip automorphisms of $$F_N$$, J. Lond. Math. Soc. (2), 91, 1, 203-224, (2015) · Zbl 1325.20035 [54] Kapovich, Ilya; Myasnikov, Alexei, Stallings foldings and subgroups of free groups, J. Algebra, 248, 2, 608-668, (2002) · Zbl 1001.20015 [55] Kapovich, Ilya; Short, Hamish, Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, Canad. J. Math., 48, 6, 1224-1244, (1996) · Zbl 0873.20025 [56] Kapovich, Michael, On the absence of Sullivan’s cusp finiteness theorem in higher dimensions. Algebra and analysis, Irkutsk, 1989, Amer. Math. Soc. Transl. Ser. 2 163, 77-89, (1995), Amer. Math. Soc., Providence, RI · Zbl 0840.57023 [57] Kapovich, Michael; Kleiner, Bruce, Hyperbolic groups with low-dimensional boundary, Ann. Sci. \'Ecole Norm. Sup. (4), 33, 5, 647-669, (2000) · Zbl 0989.20031 [58] Kent, Richard P., IV; Leininger, Christopher J., Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal., 18, 4, 1270-1325, (2008) · Zbl 1282.20046 [59] Kent, Richard P., IV; Leininger, Christopher J., Uniform convergence in the mapping class group, Ergodic Theory Dynam. Systems, 28, 4, 1177-1195, (2008) · Zbl 1153.57013 [60] Klarreich, Erica, Semiconjugacies between Kleinian group actions on the Riemann sphere, Amer. J. Math., 121, 5, 1031-1078, (1999) · Zbl 1011.30035 [61] Leininger, Christopher; Long, Darren D.; Reid, Alan W., Commensurators of finitely generated nonfree Kleinian groups, Algebr. Geom. Topol., 11, 1, 605-624, (2011) · Zbl 1237.20044 [62] Leininger, Christopher J.; Mj, Mahan; Schleimer, Saul, The universal Cannon-Thurston map and the boundary of the curve complex, Comment. Math. Helv., 86, 4, 769-816, (2011) · Zbl 1248.57003 [63] Levitt, Gilbert; Lustig, Martin, Irreducible automorphisms of $$F_n$$ have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu, 2, 1, 59-72, (2003) · Zbl 1034.20038 [64] McMullen, Curtis T., Local connectivity, Kleinian groups and geodesics on the blowup of the torus, Invent. Math., 146, 1, 35-91, (2001) · Zbl 1061.37025 [65] Minsky, Yair N., On rigidity, limit sets, and end invariants of hyperbolic $$3$$-manifolds, J. Amer. Math. Soc., 7, 3, 539-588, (1994) · Zbl 0808.30027 [66] Minsky, Yair, The classification of Kleinian surface groups. I. Models and bounds, Ann. of Math. (2), 171, 1, 1-107, (2010) · Zbl 1193.30063 [67] Mitra, M., Ending laminations for hyperbolic group extensions, Geom. Funct. Anal., 7, 2, 379-402, (1997) · Zbl 0880.57001 [68] Mitra, Mahan, Cannon-Thurston maps for hyperbolic group extensions, Topology, 37, 3, 527-538, (1998) · Zbl 0907.20038 [69] Mitra, Mahan, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom., 48, 1, 135-164, (1998) · Zbl 0906.20023 [70] [Miy] H. Miyachi, \newblockSemiconjugacies between actions of topologically tame Kleinian groups, preprint, 2002. [71] Mj, Mahan, Ending laminations and Cannon-Thurston maps, Geom. Funct. Anal., 24, 1, 297-321, (2014) · Zbl 1297.57040 [72] Mj, Mahan, Cannon-Thurston maps for pared manifolds of bounded geometry, Geom. Topol., 13, 1, 189-245, (2009) · Zbl 1166.57009 [73] Mj, Mahan, Cannon-Thurston maps, i-bounded geometry and a theorem of McMullen. Actes du S\'eminaire de Th\'eorie Spectrale et G\'eometrie. Volume 28. Ann\'ee 2009–2010, S\'emin. Th\'eor. Spectr. G\'eom. 28, 63-107, (2010), Univ. Grenoble I, Saint-Martin-d’H\`eres · Zbl 1237.57018 [74] Mj, Mahan, Cannon-Thurston maps and bounded geometry. Teichm\"uller theory and moduli problem, Ramanujan Math. Soc. Lect. Notes Ser. 10, 489-511, (2010), Ramanujan Math. Soc., Mysore · Zbl 1204.57014 [75] [M10b] M. Mj, Cannon-Thurston Maps for Kleinian Groups, arXiv:1002.0996 · Zbl 1370.57008 [76] Mj, Mahan, On discreteness of commensurators, Geom. Topol., 15, 1, 331-350, (2011) · Zbl 1209.57010 [77] Mj, Mahan, Cannon-Thurston maps for surface groups, Ann. of Math. (2), 179, 1, 1-80, (2014) · Zbl 1301.57013 [78] Mj, Mahan; Pal, Abhijit, Relative hyperbolicity, trees of spaces and Cannon-Thurston maps, Geom. Dedicata, 151, 59-78, (2011) · Zbl 1222.57013 [79] Queff\'elec, Martine, Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics 1294, xvi+351 pp., (2010), Springer-Verlag, Berlin · Zbl 1225.11001 [80] Rivin, Igor, Zariski density and genericity, Int. Math. Res. Not. IMRN, 19, 3649-3657, (2010) · Zbl 1207.20045 [81] [Sou] J. Souto, Cannon-Thurston maps for thick free groups, preprint, 2006. http://www.math.ubc.ca/$$~$$jsouto/papers/Cannon-Thurston.pdf. [82] Sullivan, Dennis, Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc. (N.S.), 6, 1, 57-73, (1982) · Zbl 0489.58027 [83] Swenson, Eric L., Quasi-convex groups of isometries of negatively curved spaces, Topology Appl., 110, 1, 119-129, (2001) · Zbl 0973.20037 [84] Tukia, P., A rigidity theorem for M\"obius groups, Invent. Math., 97, 2, 405-431, (1989) · Zbl 0674.30038 [85] Tukia, Pekka, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math., 23, 2, 157-187, (1994) · Zbl 0855.30036 [86] Tukia, Pekka, Conical limit points and uniform convergence groups, J. Reine Angew. Math., 501, 71-98, (1998) · Zbl 0909.30034 [87] [Th] W. P. Thurston, The geometry and topology of three-manifolds, Lecture Notes from Princeton University, 1978–1980. [88] Yaman, Asli, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math., 566, 41-89, (2004) · Zbl 1043.20020
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.