Wow. Almost everyone in here is wrong. These threads are rare!
Anyway I'm gonna make a comment that might be actually helpful to you personally, while trying to not be condescending or rude or whatever.
To clarify his afk'ing etc is not right , obviously. Nothing can excuse that.
However if Jax is ahead, and doing well it is almost NEVER right for him to group with you. And playing a split pusher while having a team refusing to work with you can be EXTREMELY FRUSTRATING! I want you to understand that, as you said you were telling him to group. I can promise you that is basically ALWAYS the wrong call.
If you're behind and can't win without him, but he can ALWAYS win 1v1 then it's worth it for him to split. He can win 1v1 which means at minimum he can force a 1v2 WHICH HE MIGHT EVEN WIN AS WELL!
If you're that behind and get crushed 4v4 (while he and his opponent face off) there is no chance you win 5v5 either. Most split pushers are bad team fighters, and can rarely make such a large difference.
However, you MIGHT be able to win 4v3 when he draws 2.
Alternatively, if they do only send the 1 to deal with him, and he can win out 1v1, your ONLY job becomes to not die! If you can stall 4v4 while he wins 1v1 then you win that situation by doing literally nothing except clearing waves. You don't HAVE to fight to win out there, you just need to not die, and delay enemies from rotating to the splitter (and then hard pressure something if they DO rotate.)
Basically it's up to you to work as a team to pressure the enemy at the same time. You should NEVER be saying 'hey jax please group'. But you might say 'hey jax, wait a second to pressure hard, so that if they 3v1 you, we are in position to baron or pressure an inhibitor' or whatever
Split pushing is a strategy that requires both the split pusher and the rest of the team to apply pressure at the same time! If he hard shoves and loses 4v1 it's bad, If he does that when you are in position to react to it, however its GREAT if he draws 4 bot lane while baron is clear. And then it comes on you and your team to execute.
Basically what I'm saying is that your mindset of 'we need him to group' is absolutely wrong, and will not help you in future games.
You dont need him to group. You need him to split and draw 1-2 away from their team so that you can win an unfair fight. Or you can remain even while he splits and carries.
Also pro tip you NEVER rotate to him if he gets collapsed on. Just go elsewhere and get something. If he gets caught 4v1 top lane, take bot tower or dragon as a team. If he gets caught splitting bot, take baron ASAP or a tower.
Anyway that's the general idea behind splitting pressure. If you choose to follow it or not that's on you. But from what I read, tbh, this game sounds like you had a free carry, and you refused to work with him, he got frustrated and left. Again I do not believe that to be right, but I don't think your method of trying to force your wrong playstyle is correct either.
Glhf in the future.