Möbius and sub-Möbius structures

The notion of a sub-Möbius structure is introduced, and necessary and sufficient conditions are found under which a sub-Möbius structure is a Möbius structure. It is shown that on the boundary at infinity $\partial _{\infty } Y$ of every Gromov hyperbolic space $Y$ there is a canonical sub-Möbius structure invariant under the isometries of $Y$ and such that the sub-Möbius topology on $\partial _{\infty } Y$ coincides with the standard one.