Examples of Mobius-like groups which are not Mobius groups

In this paper we give two basic constructions of groups with the following properties: (a) G hooked right arrow Homeo(+)(S-1), i.e., the group G is acting by orie ntation preserving homeomorphisms on S-1; (b) every element of G is Mobius-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 Mobius group. Both constructions have the same basic idea (inspired by Denjoy): we start with a Mobius group H (of a certain type) and then we change the underlying circle upon which H acts by inserting some closed intervals and then exten ding the group action over the new circle. We denote this new action by (H) over bar. Now we form a new group G which is generated by all of (H) over bar and an additional element g whose existence is enabled by the inserted intervals. This group G has all the properties (a) through (e).