is there an isotropy subgroup for a properly discontinuous action The article on Orbifold says The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups In particular this applies to any action of a finite group I want to agree But I m confused If the action of the group were free properly discontinuous we d get a manifold as a quotient wouldn t we But how does this make sense How can a properly discontinuous action be non free If there is a point that is fixed by some group element we ll always have that group element in the set How can that be the case And why is freely discontinuous a redirect to this article What does freely discontinuous mean This is probably terrible obvious but I don t see it right now Maybe tomorrow