The thin wire you are describing sounds like the "kill" wire for the coil (owner of the big thick wire to the spark plug). When the thin wire is grounded, it shorts power in the coil to stop the engine.
Most engines use the throttle position on the carb to ground out this wire, although some use part of the engine brake system (that irritating bale handle you have to hold to keep the engine running) to ground out the coil.
