Most group games I've seen are PBF (play by forum), which is an easy way for all players to be able to access the game files and be notified of new posts to the thread. (i.e. when someone else finishes their turn)
TripleA also has a history feature, so you can roll the game back to any point rather than having to fish around to find the file from a certain point in the game.
If you're more interested in being able to see a picture of the board (perhaps so people can view on a cell phone without loading TripleA), TripleA does allow you to export a current picture of the game. However they can be fairly large, and would require players to manually export, and attach/upload it at the end of their turn. (in Triplea go Export -> Export Gameboard Picture)