Geometric finiteness in negatively pinched Hadamard manifolds. (English) Zbl 1512.53035
Summary: In this paper, we generalize Bonahon’s characterization of geometrically infinite torsion-free discrete subgroups of \(\mathrm{PSL}(2,\mathbf C)\) to geometrically infinite discrete subgroups \(\Gamma\) of isometries of negatively pinched Hadamard manifolds \(X\). We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup \(\Gamma\) has a set of nonconical limit points with the cardinality of the continuum.
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 20F65 | Geometric group theory |
| 22E40 | Discrete subgroups of Lie groups |
| 57N16 | Geometric structures on manifolds of high or arbitrary dimension |
References:
| [1] | Ahlfors, L. V.: Fundamental polyhedrons and limit point sets of Kleinian groups. - Proc. Nat. Acad. Sci. U.S.A. 55, 1966, 251-254. · Zbl 0132.30801 |
| [2] | Ballmann, W.: Lectures on spaces of nonpositive curvature. With an appendix by Misha Brin. - DMV Seminar 25, Birkhäuser Verlag, Basel, 1995. · Zbl 0834.53003 |
| [3] | Ballmann, W., M. Gromov, and V. Schroeder: Manifolds of nonpositive curvature. Progr. Math. 61, Birkhäuser, Boston, 1985. · Zbl 0591.53001 |
| [4] | Beardon, A.: The geometry of discrete groups. - Springer-Verlag, Berlin-New York, 1983. · Zbl 0528.30001 |
| [5] | Beardon, A., and B. Maskit: Limit sets of Kleinian groups and finite sided fundamental polyhedra. - Acta Math. 132, 1974, 1-12. · Zbl 0277.30017 |
| [6] | Bishop, C. J.: On a theorem of Beardon and Maskit. - Ann. Acad. Sci. Fenn. Math. 21, 1996, 383-388. · Zbl 0849.30034 |
| [7] | Bishop, C. J., and P. W. Jones: The law of the iterated logarithm for Kleinian groups. - In: Lip’s legacy (New York, 1995), Contemp. Math. 211, 1997, 17-50. · Zbl 1126.30315 |
| [8] | Bishop, R., and R. Crittenden: Geometry of manifolds. Reprint of the 1964 original. - AMS Chelsea Publishing, Providence, RI, 2001 · Zbl 0984.53001 |
| [9] | Bonahon, F.: Bouts des varietes hyperboliques de dimension 3. - Ann. of Math. 124, 1986, 71-158. · Zbl 0671.57008 |
| [10] | Bowditch, B. H.: Geometrical finiteness for hyperbolic groups. - J. Funct. Anal. 113, 1993, 245-317. · Zbl 0789.57007 |
| [11] | Bowditch, B. H.: Geometrical finiteness with variable negative curvature. - Duke Math. J. 77:1, 1995, 229-274. · Zbl 0877.57018 |
| [12] | Breuillard, E., and K. Fujiwara: On the joint spectral radius for isometries of nonpositively curved spaces and uniform growth. - Preprint, arXiv:1804.00748. · Zbl 1539.20020 |
| [13] | Cheeger, J., and D. G. Ebin: Comparison theorems in Riemannian geometry. - NorthHolland Math. Lib. 9, North-Holland, Amsterdam, 1975. · Zbl 0309.53035 |
| [14] | Dey, S., M. Kapovich, and B. Liu: Ping-pong in Hadamard manifolds. - Preprint, arXiv:1806.07020. · Zbl 1455.53061 |
| [15] | Druţu, C., and M. Kapovich: Geometric group theory. - AMS series Colloquium Publications, 2018. · Zbl 1447.20001 |
| [16] | Falk, K., K. Matsuzaki, and B. O. Stratmann: Checking atomicity of conformal ending measures for Kleinian groups. - Conform. Geom. Dyn. 14, 2010, 167-183. · Zbl 1218.30121 |
| [17] | Fernández, J. L., and M. V. Melián: Escaping geodesics of Riemannian surfaces. - Acta Math. 187:2, 2001, 213-236. · Zbl 1001.53025 |
| [18] | Hatcher, A.: Algebraic topology. - Cambridge Univ. Press, 2002. · Zbl 1044.55001 |
| [19] | Kurosh, A.: The theory of groups. Volume 2. - Chelsea, New York, 2nd ed., 1960. · Zbl 0094.24501 |
| [20] | Marden, A.: The geometry of finitely generated Kleinian groups. - Ann. of Math. (2) 99, 1974, 383-462. · Zbl 0282.30014 |
| [21] | Montejano, L.: A new topological Helly theorem and some transversal results. - Discrete Comput. Geom. 52:2, 2014, 390-398. · Zbl 1322.52006 |
| [22] | Nicholls, P.: The ergodic theory of discrete groups. - London Math. Soc. Lecture Note Ser. 143, Cambridge Univ. Press, Cambridge, 1989. · Zbl 0674.58001 |
| [23] | Paulin, F.: On the critical exponent of a discrete group of hyperbolic isometries. - Differential Geom. Appl. 7, 1997, 231-236. · Zbl 0885.53044 |
| [24] | Ratcliffe, J.: Foundations of hyperbolic manifolds. - Grad. Texts in Math. 149, Berlin, New York, Springer-Verlag, 2nd ed., 1994. · Zbl 0809.51001 |
| [25] | Thurston, W. P.: The geometry and topology of 3-manifolds. Chapter 4. - Revised notes, 1982. · Zbl 0483.57007 |
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