Examples of Möbius-like groups which are not Möbius groups

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).