I connected an SRF05 to my picaxe, but my ground connection wire snapped off the sensor’s PCB. but I turned it on without noticing the wire was broken, and it still works fine. the sensor seems to work woth no ground connection. Has anyone else found this, and does it cause any problems?
It may be because the board
It may be because the board is using the in or out as ground as well.
But heck, if it works, it works. It will not burn anything, that is for sure 
Will try it next time, just omit the ground
I’d recommend resoldering it
I’d recommend resoldering it to ground. The pulsin/out might not have been made to handle such a load and could later cause some damage.
I only bend rules in electronics if I’m not afraid of taking the component and instantly throwing it in the trash, as could happen if it fried.
heat
I found out in school today that this may lead to overheating. it is best to solder it back on