Examples of Möbius-like groups which are not Möbius groups

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).