Examples of Möbius-like groups which are not Möbius groups

Kovačević, Nataša

doi:10.1090/S0002-9947-99-02188-1

Examples of Möbius-like groups which are not Möbius groups
HTML articles powered by AMS MathViewer

by Nataša Kovačević PDF

Trans. Amer. Math. Soc. 351 (1999), 4823-4835 Request permission

Abstract:

In this paper we give two basic constructions of groups with the following properties:

$G \hookrightarrow \operatorname {Homeo}_+(\mathcal {S}^1)$, i.e., the group $G$ is acting by orientation preserving homeomorphisms on $S^1$;

every element of $G$ is Möbius-like;

$L(G) = S^1$, where $L(G)$ denotes the limit set of $G$;

$G$ is discrete;

$G$ is not a conjugate of a Möbius group.

Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group $H$ (of a certain type) and then we change the underlying circle upon which $H$ acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by $\overline {H}$. Now we form a new group $G$ which is generated by all of $\overline {H}$ and an additional element $g$ whose existence is enabled by the inserted intervals. This group $G$ has all the properties (a) through (e).

References

Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
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.
David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
F. W. Gehring and G. J. Martin, Discrete quasiconformal groups. I, Proc. London Math. Soc. (3) 55 (1987), no. 2, 331–358. MR 896224, DOI 10.1093/plms/s3-55_{2}.331
A. Hinkkanen, Abelian and nondiscrete convergence groups on the circle, Trans. Amer. Math. Soc. 318 (1990), no. 1, 87–121. MR 1000145, DOI 10.1090/S0002-9947-1990-1000145-X
N. Kovačević, Möbius-like Groups of Homeomorphisms of the Circle, Trans. A.M.S. 351 (1999).
Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1–54. MR 961162, DOI 10.1515/crll.1988.391.1