Fetching email automatically then reading it locally

My email is kept on a remote server and accessible from anywhere by IMAP. What I'd like is a mail client that will periodically sync that remote mail to my laptop in the backgroun, so that when I open my mail client, today's email is instantly there and does not need to be downloaded. The background sync process could be some separate daemon or a function of the mail client itself. What I have now is just Thunderbird accessing the remote mail server via IMAP. But it means that mail only syncs when I start Thunderbird, and it takes ages to do so before I can read it. It's a slow process. I have a feeling that fetchmail could form part of what I want, as it can download mail to a local maildir. But then what's the easiest way to access this mail locally? And would fetchmail be smart enough to handle two-way sync, so that when I delete, move or save a mail locally the change will make its way to my remote IMAP server? Or is there some other Thunderbird-like mail client that can sync in the background (I work in Gnome), solving my problem?