You do have rights. How he gets mail delivered is his problem. He can just as easily get a POBox and have mail sent there.
The question is whether your ex-boyfriend is a sub-tenant or a guest? If he agreed to pay you rent, then he is a sub-tenant and you will have to terminate his lease, giving the required notice of that according to the landlord-tenant laws of GA.
If there was no agreement that he would pay you a specified amount of money, or an agreement that he would pay certain bills, then he is a guest. You can bring an action to have a guest removed in an action called "Unlawful Detainer".
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In addition to the answer provided by Mr. Lampert...
If your ex-boyfriend is truly abusive, you have the right to request a protective order. If one is granted (if the courts believe your explanation as to why you fear him), then he will be prevented from coming within so many miles of your residence.
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
The bigger issue here is that you indicate your ex-boyfriend is emotionally and psychologically abusive. This appears to be the real issue. Good luck sorting it out. A protective order might be granted, but you should consult an attorney that offers free consultations on such things first.
Your ex-boyfriend is wrong that you have no rights. You do have rights–especially since you're the only one with a name on the lease.
As for whether he is a sub-tenant, residential leases typically prohibit sub-leasing to sub-tenants. This means your ex-boyfriend, even if he is a sub-tenant, is likely an unauthorized, unratified sub-tenant. Therefore he would have no right of possession in the premises, although he might have a separate contract claim against you individually.
The short answer is that your lease's contents will dictate whether he has any rights. Based on the limited facts you present, it's likely he has no right of possession and, at most, has only a contract claim against you. Discussing your facts in more detail with an attorney, however, will provide better clarification of your options.
Remember, if you do kick him out, that doesn't give you the right to trash his stuff. His stuff is still his stuff, and damage to it might result in additional claims against you individually. Still, it sounds like kicking him out is where you are headed.
Good luck with it, and stay safe. These kinds of situations can be tricky.
This information DOES NOT create an attorney/client relationship. I only represent clients after I've entered into an agreement with them. This is general legal information geared towards Georgia law. If your problem is not one involving Georgia law, then you should consult an attorney in your jurisdiction. THIS IS NOT LEGAL ADVICEâ€”it is provided for informational purposes only. You should always contact an attorney if you have more questions.