Steve
It sounds like there is a problem with your wiring somewhere. All equipment grounds should get connected together and this should not cause a AFCI or GFCI to trip. Neither of these devices look for current on the equipment ground they are only looking for an imbalance on the hot and grounded (neutral) conductors. Is an equipment ground run with the GFCI circuit conductors? It sounds almost like the equipment ground is not connected in the panel but is connected to the neutral wire of the circuit somewhere.
Curt
