zbMATH — the first resource for mathematics

Limits of limit sets. I. (English) Zbl 1287.30007
The authors study the convergence of a sequence of Cannon-Thurston maps (CT-maps) defined on the limit set of a fixed Kleinian group. Let \(\Gamma\) be a geometrically finite Kleinian group. Let \((\rho_n)\) be a sequence of weakly type-preserving isomorphisms from \(\Gamma\) to geometrically finite Kleinian groups \(G_n\). The authors show that if \((G_n)\) converges strongly to a geometrically finite Kleinian group \(G_{\infty}=\rho_{\infty}(\Gamma)\), then the sequence \((\hat{i}_{n})\) of CT-maps from the limit set \(\Lambda_{\Gamma}\) to \(\Lambda_{G_n}\) converges uniformly to the CT-map \(\hat{i}_{\infty}:\Lambda_{\Gamma}\to \Lambda_{G_{\infty}}\). It is also shown that if \((G_n)\) converges algebraically to \(G_{\infty}\) and the geometrical limit of \((G_n)\) is geometrically finite, then the sequence of CT-maps converges pointwise to \(\hat{i}_{\infty}:\Lambda_{\Gamma}\to \Lambda_{G_{\infty}}\). The key to prove uniform convergence of \((\hat{i}_n)\) is to construct a certain function \(f_1:\mathbb{N}\to \mathbb{N}\) such that \(f_1(N)\to \infty\) as \(N \to \infty\) and such that, for any \(d_{\Gamma}\)-geodesic segment lying outside the ball of center 1 (the unit element) and radius \(N\) in the Cayley graph of \(\Gamma\), the corresponding \(\mathbb{H}^3\)-geodesic determined by \(\rho_n\) lies outside the ball of center \(O\) (base point) and radius \(f_1(N)\) in \(\mathbb{H}^3\) for all \(n\). For the case of pointwise convergence, a similar function which depends on each limit point is constructed.
As an application the authors give an alternative proof of the \(\lambda\)-lemma for the set of attracting fixed points of loxodromic elements in \(\Gamma\).

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI arXiv
[1] Agol, I.: Tameness of hyperbolic 3-manifolds. arXiv:math/0405568v1, (2004)
[2] Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. In: Springer Grundlehren 319, Springer, Berlin, (1999) · Zbl 0988.53001
[3] Brock, J., Iteration of mapping classes and limits of hyperbolic 3-manifolds, Inventiones Math., 1043, 523-570, (2001) · Zbl 0969.57011
[4] Calegari, D.; Gabai, D., Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Am. Math. Soc., 19, 385-446, (2006) · Zbl 1090.57010
[5] Canary, R., Ends of hyperbolic 3-manifolds, J. Am. Math. Soc., 6, 1-35, (1993) · Zbl 0810.57006
[6] Cannon, J.; Thurston, W.P., Group invariant Peano curves, Geom. Topol., 11, 1315-1355, (2007) · Zbl 1136.57009
[7] Evans, R.: Deformation spaces of hyperbolic 3-manifolds: strong convergence and tameness. Ph.D. Thesis, University of Michigan, (2000) · Zbl 1193.30063
[8] Evans, R., Weakly type-preserving sequences and strong convergence, Geometriae Dedicata, 108, 71-92, (2004) · Zbl 1077.30040
[9] Fenchel, W., Nielsen, J.: Discontinuous groups of isometries in the hyperbolic plane. In: de Gruyter Studies in Mathematics 319, Berlin (2003) · Zbl 1022.51016
[10] Floyd, W., Group completions and limit sets of Kleinian groups, Inventiones Math., 57, 205-218, (1980) · Zbl 0428.20022
[11] Francaviglia, S., Constructing equivariant maps for representations, Ann. Inst. Fourier, 59, 393-428, (2009) · Zbl 1171.57016
[12] Ghys, E., de la Harpe, P., (eds): Sur les groupes hyperboliques d’après Mikhael Gromov. In: Progress in Mathematics, vol. 83. Birkhauser, Boston, (1990) · Zbl 0731.20025
[13] Kerckhoff, S.; Thurston, W., Non-continuity of the action of the modular group at the bers’ boundary of Teichmüller space, Inventiones Math., 100, 25-48, (1990) · Zbl 0698.32014
[14] Kuusalo, T., Boundary mappings of geometric isomorphisms of Fuchsian groups, Ann. Acad. Sci. Fennicae Ser. A Math., 545, 1-7, (1973) · Zbl 0272.30023
[15] Jørgensen, T.; Marden, A., Algebraic and geometric convergence of Kleinian groups, Math. Scand., 66, 47-72, (1990) · Zbl 0738.30032
[16] Mañé, R.; Sad, P.; Sullivan, D., On the dynamics of rational maps, Ann. Sci. École. Norm. Sup., 16, 193-217, (1983) · Zbl 0524.58025
[17] Marden A.: Outer Circles: An Introduction to Hyperbolic 3-Manifolds. Cambridge University Press, Cambridge (2007) · Zbl 1149.57030
[18] McMullen, C.T., Hausdorff dimension and conformal dynamics I: strong convergence of Kleinian groups, J. Differ. Geom., 51, 471-515, (1999) · Zbl 1023.37028
[19] McMullen, C.T., Local connectivity, Kleinian groups and geodesics on the blow-up of the torus, Inventiones Math., 97, 95-127, (2001) · Zbl 0672.30017
[20] Minsky, Y.N., The classification of punctured-torus groups, Ann. Math., 149, 559-626, (1999) · Zbl 0939.30034
[21] Minsky, Y.N., The classification of Kleinian surface groups I: models and bounds, Ann. Math., 171, 1-107, (2010) · Zbl 1193.30063
[22] Mitra, M., Cannon-Thurston maps for hyperbolic group extensions, Topology, 37, 527-538, (1998) · Zbl 0907.20038
[23] Miyachi, H.: Moduli of continuity of Cannon-Thurston maps. In: Spaces of Kleinian groups; London Mathematical Society Lecture Notes 329, pp. 121-150. Cambridge University Press, (2006) · Zbl 1098.30032
[24] Mj, M.: Cannon-Thurston Maps for Kleinian Groups. preprint, arXiv:1002.0996, (2010) · Zbl 1370.57008
[25] Mj, M., Series, C.: Limits of limit sets II. In preparation · Zbl 1369.30045
[26] Nielsen, J., Untersuchungen zur topologie der geschlossenen zweiseitigen flächen, Acta Math., 50, 189-358, (1927) · JFM 53.0545.12
[27] Swarup, G.A., Two finiteness properties in 3-manifolds, Bull. Lond. Math. Soc., 12, 296-302, (1980) · Zbl 0457.57007
[28] Thurston, W., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., 50, 357-382, (1982) · Zbl 0496.57005
[29] Tukia, P., On isomorphisms of geometrically finite Möbius groups, IHES Publ., 61, 127-140, (1985) · Zbl 0572.30036
[30] Tukia, P.: A remark on a paper by Floyd. In: Holomorphic Functions and Moduli, vol. II; MSRI Publ. 11, pp. 165-172. Springer, New York (1988) · Zbl 0653.30027
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.