Are all locally compact anisotropic groupoids etale up to equivalence?

• By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the category of "open principal bi-bundles", Or equivalently the isomorphism of the associated stack for the pre-topology whose covering are open surjection .

• Locally compact mean that both the space of objects and the space of morphisms are locally compact topological sapces.

• A groupoid is said to be anisotropic if all its isotropy groups are zero (i.e. point have no non trivial automorphisms), they are also sometimes called principal groupoid.

• A Groupoid is said to be etale if its source and target map are etale maps (locale homeomorphism). They are the same as r-discrete groupoids.

All the examples of anisotropic locally compact groupoids I know also happen to be equivalent to etale groupoids.

For example, for a foliation groupoids one can construct an equivalent etale groupoid by constructing a so called "transverse groupoid" whose space of objects is a disjoint union of open of $R^{n-k}$ embedded transversally to the leaves. There is a similar construction for Lie groupoids whose isotropy groups are discrete.

I also have the impression that a a similar results holds for etale complete groupoids (but this needs to be checked...).

So my question is:

Is there any example of a locally compact anisotropic groupoid that is not equivalent to an etale groupoid ? Conversely is there some other partial results showing that such example cannot exist under some additional assumptions ?

(I would also be interested by the case where the isotropy group are non zero but discrete in the sense that the isotropy group is etale over the space of objects)

• Isn't a groupoid that is anisotropic just an equivalence relation? – David Roberts Aug 14 '15 at 3:03
• Except that you can have a topology on the relation different from the topology induced by the product topology (this is important if you want to have etale example). – Simon Henry Mar 9 '16 at 16:52
• Ah, good point! – David Roberts Mar 9 '16 at 21:27