IMAP is only a way to view messages on the server. You set it to show a particular amount of mail for example, two months. Once a mail goes past the two month period, it drops off the local machine and that's it.
POP3 only syncs the inbox.
IMAP can sync all the folders you choose, but only for a specific time period, then they disappear as they are only stored temporarily.
http://en.wikipedia.org/wiki/Internet_Message_Access_Protocol
If your user buys his own domain name and moves away from the restrictions that AOL places on him, he will find life much easier and more flexible. Currently AOL restricts access to certain websites it presumably deems unsafe for viewing. I have a Family Tree Website that AOL won't allow access to and there is nothing Adult or offensive about the site.
If I come across an AOL customer, I try to encourage them away from AOL and into the real world - full of options, rather than the restricted world they have been living in.