Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature
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- by Hiroyasu Izeki
- Proc. Amer. Math. Soc. 130 (2002), 3731-3740
- DOI: https://doi.org/10.1090/S0002-9939-02-06504-8
- Published electronically: May 14, 2002
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Abstract:
We give a sufficient condition for a higher dimensional Kleinian group $\Gamma \subset \operatorname {Isom} (\mathbb {H}^{n+1})$ to be convex cocompact in terms of the critical exponent of $\Gamma$. As a consequence, we see that the fundamental group of a compact conformally flat manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally flat manifolds with positive scalar curvature.References
- B. N. Apanasov, Conformal Geometry of Discrete Groups and Manifolds, Walter de Gruiyter, Berlin, 2000.
- Christopher J. Bishop and Peter W. Jones, Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997), no. 1, 1–39. MR 1484767, DOI 10.1007/BF02392718
- Jean-Pierre Bourguignon, Une stratification de l’espace des structures riemanniennes, Compositio Math. 30 (1975), 1–41 (French). MR 418147
- B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), no. 2, 245–317. MR 1218098, DOI 10.1006/jfan.1993.1052
- E. B. Davies, Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal. 80 (1988), no. 1, 16–32. MR 960220, DOI 10.1016/0022-1236(88)90062-6
- L. Habermann, Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures, LNM 1743, Springer-Verlag, Berlin, Heidelberg, 2000.
- Hiroyasu Izeki, Limit sets of Kleinian groups and conformally flat Riemannian manifolds, Invent. Math. 122 (1995), no. 3, 603–625. MR 1359605, DOI 10.1007/BF01231457
- Hiroyasu Izeki, Quasiconformal stability of Kleinian groups and an embedding of a space of flat conformal structures, Conform. Geom. Dyn. 4 (2000), 108–119. MR 1799652, DOI 10.1090/S1088-4173-00-00062-X
- Hiroyasu Izeki and Shin Nayatani, Canonical metric on the domain of discontinuity of a Kleinian group, Séminaire de Théorie Spectrale et Géométrie, Vol. 16, Année 1997–1998, Sémin. Théor. Spectr. Géom., vol. 16, Univ. Grenoble I, Saint-Martin-d’Hères, [1998], pp. 9–32. MR 1666506, DOI 10.5802/tsg.194
- G. G. Kasparov, $K$-theory, group $C^*$-algebras, and higher signatures (conspectus), Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 226, Cambridge Univ. Press, Cambridge, 1995, pp. 101–146. MR 1388299, DOI 10.1017/CBO9780511662676.007
- Osamu Kobayashi, Scalar curvature of a metric with unit volume, Math. Ann. 279 (1987), no. 2, 253–265. MR 919505, DOI 10.1007/BF01461722
- Jacques Lafontaine, Conformal geometry from the Riemannian viewpoint, Conformal geometry (Bonn, 1985/1986) Aspects Math., E12, Friedr. Vieweg, Braunschweig, 1988, pp. 65–92. MR 979789
- Shin Nayatani, Patterson-Sullivan measure and conformally flat metrics, Math. Z. 225 (1997), no. 1, 115–131. MR 1451336, DOI 10.1007/PL00004301
- S. J. Patterson, Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 281–323. MR 903855
- Jonathan Rosenberg, $C^{\ast }$-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 197–212 (1984). MR 720934
- R. Schoen and S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), no. 1, 47–71. MR 931204, DOI 10.1007/BF01393992
- Pekka Tukia, On isomorphisms of geometrically finite Möbius groups, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 171–214. MR 783351
Bibliographic Information
- Hiroyasu Izeki
- Affiliation: Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan
- Email: izeki@math.tohoku.ac.jp
- Received by editor(s): March 19, 2001
- Received by editor(s) in revised form: July 31, 2001
- Published electronically: May 14, 2002
- Communicated by: Ronald A. Fintushel
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 3731-3740
- MSC (1991): Primary 58H15; Secondary 53A30
- DOI: https://doi.org/10.1090/S0002-9939-02-06504-8
- MathSciNet review: 1920055