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)