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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Examples of Möbius-like groups which are
not Möbius groups


Author: Natasa Kovacevic
Journal: Trans. Amer. Math. Soc. 351 (1999), 4823-4835
MSC (1991): Primary 57S05
Published electronically: August 20, 1999
MathSciNet review: 1473446
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Abstract: In this paper we give two basic constructions of groups with the following properties:

(a)
$G \hookrightarrow {Homeo_{+}(S^{1})}$, i.e., the group $G$ is acting by orientation preserving homeomorphisms on ${S^{1}}$;
(b)
every element of $G$ is Möbius-like;
(c)
${L(G)}= {S^{1}}$, where ${L(G)}$ denotes the limit set of $G$;
(d)
$G$ is discrete;
(e)
$G$ 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).


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Additional Information

Natasa Kovacevic
Affiliation: Department of Mathematics, University of Toronto, 100 St. George Street, Room 4072, Toronto, Ontario M5S 1A1, Canada
Email: natasak@home.com

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02188-1
PII: S 0002-9947(99)02188-1
Received by editor(s): March 7, 1995
Received by editor(s) in revised form: July 31, 1997
Published electronically: August 20, 1999
Article copyright: © Copyright 1999 American Mathematical Society