Hello,
I would like to know if it is possible to sync only the rank of the player and not these permissions.

Thanks you !

Perhaps split storage is what you're looking for? (check the config)

Beyond that, no there's not any functionality like it.

I'd like to link my lobby ranks with my Faction server but not permissions.
For example: I give the permission "essentials.fly" on the lobby, the player can fly on the lobby but it does not put the permission on the faction

Ah, you can use server specific permissions for this. :)

e.g.

/lp group default permission set essentials.fly true server=lobby

More info is available here:
https://github.com/lucko/LuckPerms/wiki/Context
https://github.com/lucko/LuckPerms/wiki/Advanced-Setup