On analytic properties of deformation spaces of Kleinian groups

Shiga, Hiroshige

doi:10.1090/tran/6563

On analytic properties of deformation spaces of Kleinian groups
HTML articles powered by AMS MathViewer

by Hiroshige Shiga PDF

Trans. Amer. Math. Soc. 368 (2016), 6627-6642 Request permission

Abstract:

Let $G_0$ be a non-elementary Kleinian group. We consider the deformation space of $D(G_0)$, the space of quasiconformal deformations of $G_0$, and its complex analytic properties. We show some analytic structures of $D(G_0)$ which are improvements of results by Bers, Kra, Maskit and McMullen. In particular, we clarify that the structures for Kleinian groups with non-simply connected components are different from those for Kleinian groups without non-simply connected components.

References

Michael Beck, Yunping Jiang, Sudeb Mitra, and Hiroshige Shiga, Extending holomorphic motions and monodromy, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 1, 53–67. MR 2920423, DOI 10.5186/aasfm.2012.3713
Lipman Bers, Holomorphic families of isomorphisms of Möbius groups, J. Math. Kyoto Univ. 26 (1986), no. 1, 73–76. MR 827159, DOI 10.1215/kjm/1250520965
Clifford J. Earle, On the Carathéodory metric in Teichmüller spaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 99–103. MR 0352450
C. J. Earle, I. Kra, and S. L. Krushkal′, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 343 (1994), no. 2, 927–948. MR 1214783, DOI 10.1090/S0002-9947-1994-1214783-6
Clifford J. Earle and Albert Marden, On holomorphic families of Riemann surfaces, Conformal dynamics and hyperbolic geometry, Contemp. Math., vol. 573, Amer. Math. Soc., Providence, RI, 2012, pp. 67–97. MR 2964074, DOI 10.1090/conm/573/11413
Clifford J. Earle and Curt McMullen, Quasiconformal isotopies, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 143–154. MR 955816, DOI 10.1007/978-1-4613-9602-4_{1}2
Clifford J. Earle and Sudeb Mitra, Variation of moduli under holomorphic motions, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998) Contemp. Math., vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39–67. MR 1759669, DOI 10.1090/conm/256/03996
Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR 1052649
John Hamal Hubbard, Sur les sections analytiques de la courbe universelle de Teichmüller, Mem. Amer. Math. Soc. 4 (1976), no. 166, ix+137. MR 430321, DOI 10.1090/memo/0166
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
Yunping Jiang and Sudeb Mitra, Some applications of universal holomorphic motions, Kodai Math. J. 30 (2007), no. 1, 85–96. MR 2319079, DOI 10.2996/kmj/1175287624
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Irwin Kra, On spaces of Kleinian groups, Comment. Math. Helv. 47 (1972), 53–69. MR 306485, DOI 10.1007/BF02566788
Irwin Kra, Deformation spaces, A crash course on Kleinian groups (Lectures at a Special Session, Annual Winter Meeting, Amer. Math. Soc., San Francisco, Calif., 1974) Lecture Notes in Math., Vol. 400, Springer, Berlin, 1974, pp. 48–70. MR 0402122
Irwin Kra and Bernard Maskit, The deformation space of a Kleinian group, Amer. J. Math. 103 (1981), no. 5, 1065–1102. MR 630778, DOI 10.2307/2374258
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
Albert Marden and Howard Masur, A foliation of Teichmüller space by twist invariant disks, Math. Scand. 36 (1975), no. 2, 211–228. MR 393584, DOI 10.7146/math.scand.a-11572
Bernard Maskit, A theorem on planar covering surfaces with applications to $3$-manifolds, Ann. of Math. (2) 81 (1965), 341–355. MR 172252, DOI 10.2307/1970619
Bernard Maskit, Self-maps on Kleinian groups, Amer. J. Math. 93 (1971), 840–856. MR 291453, DOI 10.2307/2373474
Katsuhiko Matsuzaki and Masahiko Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1638795
Curtis T. McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998), no. 2, 283–320. MR 1478844, DOI 10.1090/S0894-0347-98-00259-8
Sudeb Mitra and Hiroshige Shiga, Extensions of holomorphic motions and holomorphic families of Möbius groups, Osaka J. Math. 47 (2010), no. 4, 1167–1187. MR 2791561
Robert Riley, Holomorphically parameterized families of subgroups of $\textrm {SL}(2,\textbf {C})$, Mathematika 32 (1985), no. 2, 248–264 (1986). MR 834494, DOI 10.1112/S0025579300011037
Hiroshige Shiga, On analytic and geometric properties of Teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), no. 3, 441–452. MR 766636, DOI 10.1215/kjm/1250521274
Zbigniew Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), no. 2, 347–355. MR 1037218, DOI 10.1090/S0002-9939-1991-1037218-8
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), no. 3-4, 243–260. MR 806415, DOI 10.1007/BF02392543