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 3 d case I took a def from some paper now I do not remember which and modifyed it a bit 12 09 5 Jul 2004 UTC