If a person does not voluntarily relinquish whatever legal rights he may have in a property, you will need to file a court action to have the court determines who the legal owners are.
Arguably, the former son in law contributed something because you would not have been able to buy the house without his contribution.
Perhaps he wants some financial incentive. Paying him a little bit of money may be cheaper than suing.
You should also look out for the due on sale clause on the loan documents. Unless you are refinancing, taking the former son in law's name off the property likely will trigger that due on sale clause. That is, the lender likely will have the right to demand full payment of all the outstanding loan right now.
On another hand, many lenders may not care as long as the monthly payments continue to be fully paid on time.
You should review the specific facts with your attorney to find out your legal options.
I agree with the previous answer and buying him out or court would seem to be your options. A quiet title action may or may not be successful, but will certainly be expensive. Therefore an attempt at a buyout seems like the starting spot.
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I agree with attorney Nyugen. Since your name is not on the Deed, your only recourse would be to file suit to assert your right to the property assuming you have evidence to support your claim property because the fact that his name in on the Deed and loan is evidence that he owns the property. You will need proof to show that you are not merely a tenant and that you both had some type of an agreement because he could actually call you a tenant and use that as a defense against any entitlement. It would be in your best interest to start by asking him what he wants since he contributed his credit and then consult with an attorney to give you guidance on how to move forward.