Yes, you can ride FulGaz 100% offline IF you have downloaded the ride video beforehand.
You will need to be connected to the internet to download the ride video file, but once the download is complete you can then ride that video (either right away or later) without any internet connection.
FulGaz does require an internet connection regularly update the ride library, install important updates, check your subscription status, and upload your ride data (post-ride email, sync with Strava, etc). If you ride offline, you will need to reconnect to the internet in order to upload your ride. Once connected, head to your Ride History and upload the ride.
Group rides and events require a constant internet connection. Your position on course, the relative position of other riders, and live leaderboards are continually updated back and forth from our servers during events, so a fast internet connection is highly recommended.