Inverse monoids of Möbius type

Let G be a group acting effectively on a Hausdorff space X, and let Y be an open dense subset of X. We show that the inverse monoid generated by elements of G regarded as partial functions on Y is an F-inverse monoid whose maximum group image is isomorphic to G. We also describe the monoid in terms of McAlister triples. This generalizes the results about Möbius transformations on the complex plane.