Möbius-like groups of homeomorphisms
of the circle
Trans. Amer. Math. Soc. 351 (1999), 4791-4822
August 23, 1999
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Abstract: An orientation preserving homeomorphism of is Möbius-like if it is conjugate in to a Möbius transformation. Our main result is: given a (noncyclic) group whose every element is Möbius-like, if has at least one global fixed point, then the whole group is conjugate in to a Möbius group if and only if the limit set of is all of . Moreover, we prove that if the limit set of is not all of , then after identifying some closed subintervals of to points, the induced action of is conjugate to an action of a Möbius group. Said differently, is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case is isomorphic, as a group, to a Möbius group.
This result has another interpretation. Namely, we prove that a group of orientation preserving homeomorphisms of whose every element can be conjugated to an affine map (i.e., a map of the form ) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of is the one of an affine group.
F. Bonahon, personal communication.
A. Casson and D. Jungreis, Seifert Fibered Spaces and Convergence Groups, Preprint.
A. Denjoy, Sur les curbes definies par les equations differentielles a la surface du tore, J. Math. Pures. Appl. 11 (1932), 333-375.
Gabai, Convergence groups are Fuchsian groups, Ann. of Math.
(2) 136 (1992), no. 3, 447–510. MR 1189862
W. Gehring and G.
J. Martin, Discrete quasiconformal groups. I, Proc. London
Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224
Hinkkanen, Abelian and nondiscrete convergence
groups on the circle, Trans. Amer. Math.
Soc. 318 (1990), no. 1, 87–121. MR 1000145
F.P. Ramsey, On a Problem in Formal Logic, Proc. London Math. Soc. 30 (1930), 264-286.
Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine
Angew. Math. 391 (1988), 1–54. MR 961162
- F. Bonahon, personal communication.
- A. Casson and D. Jungreis, Seifert Fibered Spaces and Convergence Groups, Preprint.
- A. Denjoy, Sur les curbes definies par les equations differentielles a la surface du tore, J. Math. Pures. Appl. 11 (1932), 333-375.
- D.Gabai, Convergence Groups are Fuchsian Groups, Ann. of Math. 136 (1992), 447-510. MR 93m:20065
- F.W. Gehring and G. Martin, Discrete Quasiconformal Groups,I, Proc. London Math. Soc. 55 (1987) 331-358. MR 88m:30057
- A. Hinkkanen, Abelian and Nondiscrete Convergence Groups on the Circle, Trans. A.M.S. 318 (1990), 87-121. MR 91g:30025
- F.P. Ramsey, On a Problem in Formal Logic, Proc. London Math. Soc. 30 (1930), 264-286.
- P. Tukia, Homeomorphic Conjugates of Fuchsian Groups, J. für Reine und Angew. Math. 391 (1988). MR 89m:30047
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Department of Mathematics, University of Toronto, 100 St. George Street, Room 4072, Toronto, Ontario M5S 1A1, Canada
Received by editor(s):
March 7, 1995
Received by editor(s) in revised form:
July 31, 1997
August 23, 1999
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