sure there aren't many such actions for general u but given an action one can choose an invariant u i do not see a problem here if you want to change def to more stadard it is fine but i like this one if you change to diffeomorphsm it will give the same orbifold but the def will become mor complecated i do not know a good ref i'm not sure but it seems that thurston was mostly interested in d case i took a def from some paper now i do not remember which and modifyed it a bit jul utc