The precision of GPS and geofences has nothing to do with internet connections. You could send a data payload to your app to be used offline, accessed via geofencing, and you wouldn't lose any precision.
A geofence is just a shape on a map, which is a latitude x longitude grid. If a device knows A) the shape of the geofence, and B) its own location, it will accurately determine whether it's within a geofence.
Neither of those things (A or B) require an internet connection. GPS operates independently from the internet. It's a one-way transmission (satellites -> device).
It's possible to detect location more quickly or more accurately using other sources to augment GPS. These include WiFi SSIDs, as mentioned above. That still doesn't require an internet connection -- only a WiFi adapter.