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In this paper we give two basic constructions of groups with the following properties: (a) G ↪→ Homeo+(S), i.e., the group G is acting by orientation preserving homeomorphisms on S1; (b) every element of G is Möbius-like; (c) L(G) = S1, 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 H. Now we form a new group G which is generated by all of H and an additional element g whose existence is enabled by the inserted intervals. This group G has all the properties (a) through (e).

Examples of Möbius-like groups which are not Möbius groups