Quasiconformally Instable Disc Bundles with Complex Structures

B. N. Apanasov

Informatics and Automation, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 18-30.

Résumé

We discuss deformations and the quasiconformal instability of the Kähler geometry of disc bundles that are locally modeled on symmetric rank-one manifolds. The Kähler geometry of these manifolds is associated with natural complex or hypercomplex structures of pinched negative sectional curvature and infinite volume. Their fundamental groups are isomorphic to discrete subgroups of $\mathrm {PU}(n,1)$, $\mathrm {PSp}(n,1)$, or $\mathrm F_4^{-20}$.

Bibliographie

[1] Apanasov B.N., “Nontriviality of Teichmüller space for Kleinian group in space”, Riemann surfaces and related topics, Proc. Conf. (Stony Brook), Ann. Math. Stud., 97, eds. I. Kra, B. Maskit, Princeton Univ. Press, Princeton, 1978, 21–31 | MR

[2] Apanasov B.N., Conformal geometry of discrete groups and manifolds, De Gruyter Expos. Math., 32, W. de Gruyter, Berlin; New York, 2000 | MR

[3] Apanasov B.N., “Nonstandard uniformized conformal structures on hyperbolic manifolds”, Invent. math., 105 (1991), 137–152 | DOI | MR | Zbl

[4] Apanasov B.N., “Kvazisimmetrichnye uzly i prostranstva Teikhmyullera”, DAN, 354:1 (1997), 7–10 | MR | Zbl

[5] Apanasov B., “Bending deformations of complex hyperbolic surfaces”, J. reine und angew. Math., 492 (1997), 75–91 | MR | Zbl

[6] Apanasov B.N., “Kanonicheskie gomeomorfizmy v gruppe Geizenberga, indutsirovannye izomorfizmami diskretnykh podgrupp v $PU(n,1)$”, DAN, 356:5 (1997), 727–730 | MR | Zbl

[7] Apanasov B.N., “Geometriya i topologiya kompleksnykh giperbolicheskikh mnogoobrazii i mnogoobrazii Koshi–Rimana”, UMN, 52:5 (1997), 9–41

[8] Apanasov B., Deformations and stability in complex hyperbolic geometry, MSRI Preprint 1997-111, Math. Sci. Res. Inst., Berkeley, 1997; http://www.msri.org/publications/preprints/online/1997-111.html

[9] Apanasov B., “Complex hyperbolic manifolds: rigidity versus flexibility and instability of deformations”, Geometric structures on manifolds, Proc. Conf. (Seoul, 1997), Lect. Notes Ser., 46, Seoul Nat. Univ., Seoul, 1999, 1–35 | MR | Zbl

[10] Apanasov B., “Deformations and rigidity of group representations in rank one symmetric spaces”, Groups–Korea'98, W. de Gruyter, Berlin; New York, 2000, 19–38 | MR | Zbl

[11] Apanasov B.N., “Deistvie nesuperzhestkikh reshetok v simmetricheskikh prostranstvakh ranga 1”, DAN, 374:5 (2000), 586–589 | MR | Zbl

[12] Apanasov B., Kim I., Numerical invariants and deformations in quaternionic and octonionic hyperbolic spaces, Prépubl. d'Orsay 98-69, Univ. de Paris-Sud, 1998, 43 pp.

[13] Apanasov B., Kim I., Rigidity of quaternionic and octonionic manifolds, Preprint, Univ. Oklahoma, 1999 | Zbl

[14] Apanasov B., Kim I., Cartan angular invariant and deformations in rank one symmetric spaces, Preprint, Univ. Oklahoma, 2001

[15] Apanasov B., Xiangdong Xie, “Discrete actions on nilpotent groups and negatively curved spaces”, Diff. Geom. and Appl., 20 (2004), 11–29 | DOI | MR | Zbl

[16] Corlette K., “Archimedean superrigidity and hyperbolic geometry”, Ann. Math., 135 (1992), 165–182 | DOI | MR | Zbl

[17] Falbel E., Koseleff P.-V., “A circle of modular groups in $\mathrm{PU}(2,1)$”, Math. Res. Lett., 9 (2002), 379–391 | MR | Zbl

[18] Falbel E., Zocca V., “A Poincaré's polyhedron theorem for complex hyperbolic geometry”, J. reine und angew. Math., 516 (1999), 133–158 | MR | Zbl

[19] Goldman W., Complex hyperbolic geometry, Oxford Math. Monogr., Clarendon Press, Oxford, 1999 | MR | Zbl

[20] Goldman W., Millson J., “Local rigidity of discrete groups acting on complex hyperbolic space”, Invent. math., 88 (1987), 495–520 | DOI | MR | Zbl

[21] Gromov M., “Volume and bounded cohomology”, Publ. Math. IHES, 56 (1982), 5–99 | MR | Zbl

[22] Gromov M., Schoen R., “Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one”, Publ. Math. IHES, 76 (1992), 165–246 | MR | Zbl

[23] Millington M.H., “Subgroups of the classical modular group”, J. London Math. Soc. Ser. 2, 1 (1969), 351–357 | DOI | MR | Zbl

[24] Miner R., “Quasiconformal equivalence of spherical CR manifolds”, Ann. Acad. sci. Fenn. AI: Math., 19 (1994), 83–93 | MR | Zbl

[25] Mostow G.D., Strong rigidity of locally symmetric spaces, Princeton Univ. Press, Princeton, 1973 | MR | Zbl

[26] Pansu P., “Métriques de Carnot–Carathéodory et quasiisométries des espaces symmétries de rang un”, Ann. Math., 129 (1989), 1–60 | DOI | MR | Zbl

[27] Toledo D., “Representations of surface groups on complex hyperbolic space”, J. Diff. Geom., 29 (1989), 125–133 | MR | Zbl