I've set up a few email accounts in a hostgator shared hosting account. The emails are accessed through gmail's IMAP service. The accounts are now over their quota and I want to free some space by deleting emails. Will emails deleted from the server get deleted from the user's gmail account? If so, what's a good solution?
On IMAP, mails remain on server. Any client essentially behaves as a remote display. So, yes, when you delete mail from IMAP, it will get deleted from server, destroying the mail completely. From your question, if I dont get it wrong, you dont want this to happen. Options that come to mind are:
You can have your clients have move those mails to a local folder in a mail client such as Thunderbird. This will create a local copy of the mail and remove it from server, freeing up space.
You can instead use a mail client and connect using POP3, which will download mails to their local computers, removing them from server.
You can purchase more space from GMail.