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Quasiconformal groups with small dilatation. I. (English) Zbl 0984.30023
Quasiconformal Fuchsian groups with small dilatation are studied in this paper. A Jørgensen-type inequality in all dimensions is established for such a class of groups. It is shown that discreteness persists to the limit under algebraic convergence. Such groups are discrete if and only if every two generator subgroup is discrete.
Reviewer: Li Zhong (Beijing)

##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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##### References:
 [1] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. · Zbl 0528.30001 [2] P. Bonfert-Taylor, Jørgensen’s inequality for discrete convergence groups, Ann. Acad. Sci. Fenn. Math. 25 (2000) 131-150. · Zbl 0945.30035 [3] C. Cao and P. L. Waterman, Conjugacy invariants of Möbius groups, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) Springer, New York, 1998, pp. 109 – 139. · Zbl 0894.30027 [4] Shmuel Friedland and Sa’ar Hersonsky, Jorgensen’s inequality for discrete groups in normed algebras, Duke Math. J. 69 (1993), no. 3, 593 – 614. · Zbl 0799.30033 · doi:10.1215/S0012-7094-93-06924-4 · doi.org [5] Michael H. Freedman and Richard Skora, Strange actions of groups on spheres, J. Differential Geom. 25 (1987), no. 1, 75 – 98. · Zbl 0588.57024 [6] F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353 – 393. · Zbl 0113.05805 [7] F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331 – 358. · Zbl 0628.30027 · doi:10.1093/plms/s3-55_2.331 · doi.org [8] Sa’ar Hersonsky, A generalization of the Shimizu-Leutbecher and Jørgensen inequalities to Möbius transformations in \?^\?, Proc. Amer. Math. Soc. 121 (1994), no. 1, 209 – 215. · Zbl 0812.30017 [9] T. Jørgenson, On discrete groups of Möbius transformations, Amer. J. Math. 96 (1976) 739-749. · Zbl 0336.30007 [10] Gaven J. Martin, Discrete quasiconformal groups that are not the quasiconformal conjugates of Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 179 – 202. · Zbl 0635.30021 · doi:10.5186/aasfm.1986.1113 · doi.org [11] G. J. Martin, On discrete Möbius groups in all dimensions: a generalization of Jørgensen’s inequality, Acta Math. 163 (1989), no. 3-4, 253 – 289. · Zbl 0698.20037 · doi:10.1007/BF02392737 · doi.org [12] Gaven J. Martin, Quasiconformal and affine groups, J. Differential Geom. 29 (1989), no. 2, 427 – 448. · Zbl 0668.53004 [13] G.J. Martin, Algebraic convergence of discrete isometry groups of negative curvature, Pacific J. Math. 160 (1992) 109-127. · Zbl 0822.57026 [14] M. Jean McKemie, Quasiconformal groups with small dilatation, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987), no. 1, 95 – 118. · Zbl 0588.30050 · doi:10.5186/aasfm.1987.1223 · doi.org [15] Dennis Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465 – 496. [16] Pekka Tukia, On two-dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 73 – 78. · Zbl 0411.30038 · doi:10.5186/aasfm.1980.0530 · doi.org [17] P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. AI Math. 10 (1985) 561-562. · Zbl 0533.30019 [18] Pekka Tukia, Convergence groups and Gromov’s metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157 – 187. · Zbl 0855.30036 [19] Jussi Väisälä, Lectures on \?-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. · Zbl 0221.30031 [20] P. L. Waterman, Möbius transformations in several dimensions, Adv. Math. 101 (1993), no. 1, 87 – 113. · Zbl 0793.15019 · doi:10.1006/aima.1993.1043 · doi.org
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