Here is what I understand from your question. 1) You were in a relationship with a man and had a child while in that relationship. 2) You were living with the man during and after having the child. 3) The father's name is on the birth certificate of the child. 4) At some point the relationship became abusive and under the advice of the DA, Victims Advocate and other individuals, you left the state. (I am assuming that the father was under investigation for a crime related to the a buse.)
From what you state, there is no current custody matter, only that you are afraid at some point in the future the child's father will seek it. You are right to be concerned because the father does have rights that have not been extinguished or limited, which can only be done through a court. You should see a family law attorney on the matter. If you can afford one, see if there are any clinics or pro-bono attorneys who might assist you in this. The New Mexico Bar Association lists a number of places where you might get help: http://www.nmbar.org/Public/legalserviceproviders.html
I am licensed in New Mexico and Pennsylvania, and therefore any discussion of issues related to other states must considered within that context. In addition, my comments are not intended to create a legal representation but merely to respond to the limited facts presented by the question. Any opinion herein is not meant as a precise statement of legal rights or as a recommendation of any particular course of action. A more complete legal review can be obtained through local counsel.
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